Dimensionless sensitivity

We now define the dimensionless sensitivity, i, of the overall rate with respect to k as... 

R dimensionless sensitivity of contact resistance to pressure = mkdRcldPa... 

In this work, we determine constraints on the dimensionless parameters of the system (dimensionless electrode widths, gap size and Peclet number), first qualitatively and then quantitatively, which ensure that the proposed flow reconstmction approach is sufficiently sensitive to the shape of the flow profile. The results can be readily applied for identification of hydrodynamic regimes or electrode geometries that provide best performance of our flow reconstmction method. 

Thermal diffusivity Temperature sensitivity Temperature difference Thickness of tube Aspect ratio, relation of Cp/Cy Fluid dielectric constant Wall zeta potential Dimensionless temperature Friction factor, Debye length Mean free path Dynamic viscosity Kinematic viscosity Bejan number Density... 

Semi-continuous reactor 423, 426, 475, 518 Semi-dimensionless form 649 Sensitivity... 

The simulation result (Figure 4) shows that when two initial conditions are very close, after a dimensionless time of 40 units the concentration of reactant A and the reactor temperature are completely different. This means that the system has a chaotic behavior and their d3mamical states diverge from each other very quickly, i.e. the system has high sensitivity to initial conditions. This separation increases with time and the exponential divergence of adjacent phase points has a very important consequence for the chaotic attractor, i.e. 

For each cathodic stripping mechanism, the dimensionless net peak current is proportional to the amount of the deposited salt, which is formed in the course of the deposition step. The amount of the salt is affected by the accumulation time, concentration of the reacting ligand, and accumulation potential. The amount of the deposited salt depends sigmoidally on the deposition potential, with a half-wave potential being sensitive to the accumulation time. If the accumulation potential is significantly more positive than the peak potential, the surface concentration of the insoluble salt is independent on the deposition potential. The formation of the salt is controlled by the diffusion of the ligand, thus the net peak current is proportional to the square root of the accumulation time. If reaction (2.204) is electrochemically reversible, the real net peak current depends linearly on the frequency, which is a common feature of all electrode mechanism of an immobilized reactant (Sect. 2.6.1). The net peak potential for a reversible reaction (2.204) is a hnear function of the log(/) with a slope equal to typical theoretical response... 

The voltammetric features of a reversible reaction are mainly controlled by the thickness parameter A = The dimensionless net peak current depends sigmoidally on log(A), within the interval —0.2 < log(A) <0.1 the dimensionless net peak current increases linearly with A. For log(A )< —0.5 the diSusion exhibits no effect to the response, and the behavior of the system is similar to the surface electrode reaction (Sect. 2.5.1), whereas for log(A) > 0.2, the thickness of the layer is larger than the diffusion layer and the reaction occurs under semi-infinite diffusion conditions. In Fig. 2.93 is shown the typical voltammetric response of a reversible reaction in a film having a thickness parameter A = 0.632, which corresponds to L = 2 pm, / = 100 Hz, and Z) = 1 x 10 cm s . Both the forward and backward components of the response are bell-shaped curves. On the contrary, for a reversible reaction imder semi-infinite diffusion condition, the current components have the common non-zero hmiting current (see Figs. 2.1 and 2.5). Furthermore, the peak potentials as well as the absolute values of peak currents of both the forward and backward components are virtually identical. The relationship between the real net peak current and the frequency depends on the thickness of the film. For Z, > 10 pm and D= x 10 cm s tlie real net peak current depends linearly on the square-root of the frequency, over the frequency interval from 10 to 1000 Hz, whereas for L <2 pm the dependence deviates from linearity. The peak current ratio of the forward and backward components is sensitive to the frequency. For instance, it varies from 1.19 to 1.45 over the frequency interval 10 < //Hz < 1000, which is valid for Z < 10 pm and Z) = 1 x 10 cm s It is important to emphasize that the frequency has no influence upon the peak potential of all components of the response and their values are virtually identical with the formal potential of the redox system. 

Mass spectrometry is a sensitive analytical technique which is able to quantify known analytes and to identify unknown molecules at the picomoles or femto-moles level. A fundamental requirement is that atoms or molecules are ionized and analyzed as gas phase ions which are characterized by their mass (m) and charge (z). A mass spectrometer is an instrument which measures precisely the abundance of molecules which have been converted to ions. In a mass spectrum m/z is used as the dimensionless quantity that is an independent variable. There is still some ambiguity how the x-axis of the mass spectrum should be defined. Mass to charge ratio should not lo longer be used because the quantity measured is not the quotient of the ion s mass to its electric charge. Also, the use of the Thomson unit (Th) is considered obsolete [15, 16]. Typically, a mass spectrometer is formed by the following components (i) a sample introduction device (direct probe inlet, liquid interface), (ii) a source to produce ions, (iii) one or several mass analyzers, (iv) a detector to measure the abundance of ions, (v) a computerized system for data treatment (Fig. 1.1). 

Once the equations of error are defined, weighting factors , in terms of estimated accuracy of the experiment, are included. These are either taken as the accuracy of the corresponding measurement as given by the original author or estimated by the person who is doing the optimisation. The error is then divided by these weighting factors to provide a dimensionless relative error for all types of experimental measurement. In addition to this, the sensitivity to the measured value of changes in temperature and composition are considered. 

Here, K is sometimes referred to as the consistency index and has units that depend on the value of the power law index, n—for example, N-s"/m. The power law index is itself dimensionless. Typical values of K and n are listed in Table 4.4. In general, the power law index is independent of both temperature and concentration, although fluids tend to become more Newtonian (n approaches 1.0) as temperature increases and concentration decreases. The consistency factor, however, is more sensitive to temperature and concentration. To correct for temperature, the following relationship is often used ... 

We now consider the dependence of the stationary-state solution on the parameter d. To represent a given stationary-state solution we can take the dimensionless temperature excess at the middle of the slab, 0ss(p = 0) or 60,ss-With the above boundary conditions, two different qualitative forms for the stationary-state locus 0O,SS — <5 are possible. If y and a are sufficiently small (generally both significantly less than i), multiplicity is a feature of the system, with ignition on increasing <5 and extinction at low <5. For larger values of a or y, corresponding to weakly exothermic processes or those with low temperature sensitivity, the hysteresis loop becomes unfolded to provide... 

They also establish the interesting fact that the dimensionless quantity (c—u)/D is much more sensitive to small changes in the gas state than D, u, P, or p. Here c is the local sound velocity. Their results are summarized in Fig 8, whose abcissa (d/a) is diameter/cell size. The cell size refers to inhomogeneities (cells) in the structure of the detonation front. These become smaller as initial pressure increases, ie, d/a is generally large at ambient pressures of 1 atm or greater... 

The SWV and SWVC curves for the EEC reaction schemes, calculated from Eqs. (7.154) and (7.157), can be seen in Figs. 7.59 and 7.60, respectively. These curves have been obtained for different values of the dimensionless chemical rate constant kc and A(i.e., the difference between the formal potentials of the electron transfer reactions). SWV proves highly sensitive for detecting the presence of the catalysis, since no measurable response is obtained in the absence of the same. From Fig. 7.59, it can be deduced that when catalysis takes place, route... 

A more comprehensive approach consists of studying the variation of the Semenov criterion as a function of the reaction energy. Such an approach is presented in [12], where the reciprocal Semenov criterion is studied as a function of the dimensionless adiabatic temperature rise. This leads to a stability diagram similar to those presented in Figure 5.2 [11, 13]. The lines separating the area of parametric sensitivity, where runaway may occur, from the area of stability is not a sharp border line it depends on the models used by the different authors. For safe behavior, the ratio of cooling rate over heat release rate must be higher than the potential of the reaction, evaluated as the dimensionless adiabatic temperature rise. 

The table below lists some common spectral interferences that are encountered in inductively coupled plasma mass spectrometry (ICP-MS), as well as the resolution that is necessary to analyze them.1 The resolution is presented as a dimensionless ratio. As an example, the relative molecular mass (RMM) of the polyatomic ion 15N160+would be 15.000108 + 15.994915 = 30.995023. This would interfere with 31P at a mass of 30.973762. The required resolution would be RMM/8RMM, or 30.973762/0.021261 = 1457. One should bear in mind that as resolution increases, the sensitivity decreases with subsequent effects on the price of the instrument. Note that small differences exist in the published exact masses of isotopes, but for the calculation of the required resolution, these differences are trivial. Moreover, recent instrumentation has provided rapid, high-resolution mass spectra with an uncertainty of less than 0.01%. 

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