Drops characteristic time scale

Unfolded state a is characterized by a wide uni-modal and asymmetrical distribution of R, with a slow rise and a sharp drop. The distribution becomes narrower and more symmetrical when the averaging time window is larger than the characteristic time scale of conformational dynamics in this unfolded state. This indicates a correlation between structure and dynamics, where conformations that correspond to shorter R move on faster time scales. 

Figure 2-15. Photographs of the relaxation of a pair of initially deformed viscous drops back to a sphere under the action of surface tension. The characteristic time scale for this surface-tension-driven flow is tc = fiRi 1 + X)/y. The properties of the drop on the left-hand side are X = 0.19, /id = 5.5 Pa s, ji = 29.3 Pa s, y = 4.4 mN/rn, R = 187 /an, and this gives tc = 1.48 s. For the drop on the right-hand side, X = 6.8, lid = 199 Pa s, //. = 29.3 Pa s, y = 4.96 mN/m, R = 217 /an, and tc = 9.99 s. The photos were taken at the times shown in the figure. When compared with the characteristic time scales these correspond to exactly equal dimensionless times (/ = t/tc) (a) t = 0.0, (b) t = 0.36, (c) t = 0.9, (d) t = 1.85, (e) t = 6.5. It will be noted that the drop shapes are virtually identical when compared at the same characteristic times. This is a first illustration of the principle of dynamic similarity, which will be discussed at length in subsequent chapters.

We denote the characteristic length and velocity scales as tc and uc, the characteristic time scale as lc/uc> and the characteristic stress and pressure scales as fj,uc/ic and X luc/Ic for the fluid outside and inside the drop, respectively. We express the interfacial tension in the form... 

Here t = (ve/so) is the characteristic time scale in a turbulent flow with specific energy dissipation so, W — AnfiRlrio/ i is the volume concentration of drops of type 2 in the flow, C is the parameter whose value depends on the chosen model of drop coagulation in a turbulent flow. 

Electrowetting, Fig. 6 (a) Transient drop and film evolution profiles for electric Bond number B = 100 for five equal time steps up to tjT = 0.5, where T = LjU is the characteristic time scale with V = being the... 

During adsorption a reduction of total pressure in the pore system of the adsorbent takes place. Besides operational and system parameters such as adsorptive concentration and adsorption equilibrium the effect is influenced by pore structure characteristics. It was found that it is less pronounced for adsorbent material with large transport pores. This is due to a higher intraparticle mass transport rate and thus a faster equilibration of the intrapafticle total pressure drop. The maximum reduction of intraparticle total pressure is obtsdned at the initial operation period. It is therefore most crucial for adsorption processes with small characteristic time scales. 

In a further analysis, the same characteristic time scale analysis was performed as before, but with the inclusion of gas-phase reactions. The calculated characteristic chemical time scales are presented in Fig. 8.4. Important intermediate species of lean CH4/air combustion such as formaldehyde (CH2O) and acetylene (C2H2) now impose further restrictions on the required time step At, as they have characteristic times of the same magnitude as total oxidation products CO2 and H2O. However, a significant drop of the chemical times to below 50 ms is evident in Fig. 8.4 for all species at — 850 K. This further justifies the use of the aforementioned combination of At and Tj for all subsequent numerical simulations. It is finally emphasized that the analysis in Figs. 8.3 and 8.4 is quite strict when applied to the channel in Fig. 8.1, since as the solid starts heating above the initial temperature T x, t — 0) — 850 K, the chemical time scales shorten substantially already at 900 K, the chemical times are a factor of 2.2-3.5 shorter than the ones at 850 K shown in Figs. 8.3 and 8.4. 

We have assumed that transfer of surfactant molecules onto the hydrophobic solid interface takes place only from the liquid-vapor interface. It is difficult to assess the contribution of surfactant molecule transfer along the solid surface from beneath the liquid. However, experimental data presented in the following text in this section support our assumption (although they do not prove it decisively). The drop surface coverage, H, is an increasing function of the bulk surfactant concentration inside the drop, whose maximum is reached close to the CMC. It follows from Equation 5.83 that at low surfactant concentrations inside the drop, the characteristic time scale of the surfactant molecules transfer, Xj, decreases with increased concentration, whereas above the CMC, Xj levels off and reaches its lowest value. Both of these effects are observed in experimental results in the following discussion (compare Figure 5.20). 

Ap Frictional pressure drop Pa CO Characteristic frequency or reciprocal time scale of flow 1/s... 

Table 1.5 Dependence of the number of micro channels N, their length L, the cross-sectional area of the reactor S and the pressure drop AP on the micro-channel diameter, when the efficiency (i.e. a fixed number of transfer units) and at least one specific characteristic quantity are kept fixed in each line. Three cases with operation time-scales varying as (c/m)°. are considered [114],...

But the entire conception here is that of equilibrium solvation of the transition state by the Debye ionic atmosphere, and closer inspection [51] indicates that this assumption can hardly be justified indeed, time scale considerations reveal that it will nearly always be violated. The characteristic time for the system to cross the reaction barrier is cot, 0.1 ps say. On the other hand, the time required for equilibration of the atmosphere is something like the time for an ion to diffuse over the atmosphere dimension, the Debye length K- this time is = 1 ns for a salt concentration C= 0.1M and only drops to lOps for C 1M. Thus the ionic atmosphere is perforce out of equilibrium during the barrier passage, and in analogy with ionic transport problems, there should be an ionic atmosphere friction operative on the reaction coordinate which can influence the reaction rate. 

As a particular example, one can consider the homogeneous nucleation in the pure water vapor at 25° C. The surface tension coefficient of water is a = 71.96 N/m at this temperature. Table 5.1 shows some characteristics of the new phase. When the oversaturation is p/p =8.1, the critical nucleus of 0.5 nm radius is seen to comprise 18 water molecules. The equihbrium pressure of such nuclei is not high (approximately 10 bar). Since the water vapor pressure in real clouds is usually no more than 0.1% over that of the saturated vapor, it is unrealistic to expect in the rea sonable time scale the homogeneous formation of water drops in Earth s atmosphere. 

Rheology is concerned with the flow and/or deformation of matter under the influence of externally imposed mechanical forces. Two limiting types of behaviour arc possible. The deformation may reverse spontaneously (relax) when the external force is removed this is called elastic behaviour and is exhibited by rigid solids. The energy used in causing the deformation is stored, and then recovered when the solid relaxes. At the other extreme, matter flows and the flow ceases (but is not reversed) when the force is removed this is called viscous behaviour and is characteristic of simple liquids. The energy needed to maintain the flow is dissipated as heat. Between the two extremes arc systems whose response to an applied force depends on the lime-scale involved. Thus pitch behaves as an elastic solid if struck but flows if left for years on a slope. Similarly, a ball of Funny Putty , a form of silicone rubber, bounces when dropped on a hard surface, when the contact time is a few milliseconds, but flows if deformed slowly on a time-scale of seconds or minutes. Systems of this kind are said to be visco-elastic. The precise nature of the observable phenomena depends on the ratio of the time it takes for the system to relax to the time taken to make an observation. This ratio is called the Deborah number (De) ... 

The estimation of Rav for characteristic parameter values shows that Rav where Aq = d/Re /" is the internal scale of turbulence. In a turbulent flow, both heat and mass exchange of drops with the gas are intensified, as compared to a quiescent medium. The delivery of substance and heat to or from the drop surface occurs via the mechanisms of turbulent diffusion and heat conductivity. The estimation of characteristic times of both processes, with the use of expressions for transport factors in a turbulent flow, has shown that in our case of small liquid phase volume concentrations, the heat equilibrium is established faster then the concentration equilibrium. In this context, it is possible to neglect the difference of gas and liquid temperatures, and to consider the temperatures of the drops and the gas to be equal. Let us keep all previously made assumptions, and in addition to these, assume that initially all drops have the same radius (21.24). Then the mass-exchange process for the considered drop is described by the same equations as before, in which the molar fluxes of components at the drop surface will be given by the appropriate expressions for diffusion fluxes as applied to particles suspended in a turbulent flow (see Section 16.2). In dimensionless variables (the bottom index 0" denotes a paramenter value at the initial conditions). 

Polymers behave as liquids when temperature changes occur at a slower rate than that required by the molecules to readjust to their new equilibrium condition. As the annealing temperature, Ta, drops farther away from Tg [i.e., (Tg — Ta) increases], the aging process slows down and the time scales involved become quite long. A temperature is eventually reached at which the characteristic rate of motion is too slow compared to the rate of temperature changes Molecular conformations are fixed, and the material is no longer able to attain stmctural equilibrium (i.e., it behaves as a glass). 

This process takes place in micron scale, but as a matter of fact, the efficiency of mixing by diffusion is length dependent, and mixing time drops significantly as diffusion distance falls below 100 pm, while in reactors with a characteristic length scale larger than 1 mm, mixing by diffusion is not very efficient. 

An explanation of the discrepancies was offered by the results from some experiments made by Johannsson et al. [30], employing a more long lived probe, Crfbpy) (t 25 /is), in the AOT-alkane-water system. From the observation of two decay processes, well separated in time as shown in Fig. 7, the authors concluded that small clusters of reverse micelles were present in the microemulsions. The initial fast process, the intramicellar quenching, occurs on the submicrosecond time scale and appears only as an initial drop since it is not resolved on the time scale used with Cr(bpy)3. It is this part of the deactivation that is possible to monitor in normal TRLQ measurements with short-lived probes. The initial drop is followed by a second decay with a characteristic time of a few microseconds before the final, very slow deactivation occurs. The results suggest that the fast exchange... 

Various experimental methods for dynamic surface tension measurements are available. Their operational time scales cover different time intervals [64,65]. Methods with a shorter characteristic operational time are the oscillating jet method [66-68], the oscillating bubble method [69-72], the fast-formed drop technique [73,74], the surface wave techniques [75-78], and the maximum bubble pressme method (MBPM) [57,79-84]. Methods of longer characteristic operational time are the inclined plate method [85], the drop-weight/volume techniques [86-90], the funnel [91] and overflowing cylinder [60,92] methods, and the axisymmetric drop shape analysis (ADSA) [93,94] see Refs. [64,65,95] for a more detailed review. 

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