Time-dependent behavior

Once in a while, polymer systems will appear to be thixotropic or rheopectic. Careful checking (including before and after molecular weight determinations) invariably shows that the phenomenon is not reversible and is due to degradation or crossUnking of the polymer when in the viscometer for long periods of time, particularly at elevated temperatures. Other transient time-dependent 

Characterization of Transformation-Toughened Ceramics Jeffrey J. Swab (U.S. Army Materials Technology Laboratory) 

To evaluate and characterize toughened ceramic matrix composite (CMC) materials for potential high temperature structural applications. At the present time four-point flexure testing is being used to determine the high temperature performance of these CMCs, However, a related in-house program to develop a tensile test for CMC materials is underway. Once this test has been developed and refined both techniques will be used to provide a more comprehensive characterization of CMC materials at elevated temperatures. 

Flexure Testing of CMC materials reinforced with continuous fibers clear limitations as pointed out by a number of authors. However, flexure testing can provide 

In order to insure tensile failure during flexure testing it has been shown that a long flexure span relative to the specimen thickness is necessary. Previous work d has recommended a ratio of at least 20. Although flexure testing has been shown to be more than adequate to examine this class of materials, direct tension testing would still be required to provide comprehensive structural characterization for design purposes. 

Failure appeared to initiate in a tensile manner then half way through the bar, approximately were the neutral axis is located, it appears to shift to a shear failure. Some of the failures initiated where fibers were exposed on the surface or at channels left behind when the fibers were removed during machining. The amount and length of fiber pull-out was much less than at room temperature. This would indicate that at these temperatures there is some increase in the fiber/matrix bond strength which would account for the reduction in fiber pull-out and the tensile type failure. 

For some fluids the relation between the shear rate and the shear stress depends on the duration of the flow, or, more precisely, on the kinematic history of the fluid. We can distinguish two categories thixotropy and rheopexy. 

FIGURE 17.7 Rheological classification of solids (a) linear elastic, (b) nonlinear elastic, and (c) viscoelastic behavior. 

Like plastic systems, a thixotropic system is transferred from a gel to a flowing system by exerting a stress o o, the yield stress. This may be achieved by simply shaking or stirring. In contrast to plastic behavior, after releasing the stress, thixotropic systems need a finite time to recover the gel state. 

Many food, cosmetic, and pharmaceutical products are thixotropic. An example where thixotropy is required is paint. The paint should be fluid when applied under stress and thereafter it must become solid but a certain recovery time is desired to let the paint flow to form a homogeneous even layer. 

When shear-thickening systems need some time to reestablish o(dy/dt) equilibrium, they are called rheopectic. Thus, rheopexy is the relatively slow increase in due to gradual interaction and structure formation. Rheopexy is a rather rare and not a very relevant phenomenon. 

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Mies F H and Krauss M 1966 Time-dependent behavior of activated molecules. High-pressure unimolecular rate constant and mass spectra J. Cham. Phys. 45 4455-68... 

Molecular dynamics is a simulation of the time-dependent behavior of a molecular system, such as vibrational motion or Brownian motion. It requires a way to compute the energy of the system, most often using a molecular mechanics calculation. This energy expression is used to compute the forces on the atoms for any given geometry. The steps in a molecular dynamics simulation of an equilibrium system are as follows ... 

The simplest model of time-dependent behavior of a neutron population in a reactor consists of the point kinetics differential equations, where the space-dependence of neutrons is disregarded. The safety of reactors is greatly enhanced inherently by the existence of delayed neutrons, which come from radioactive decay rather than fission. The differential equations for the neutron population, n, and delayed neutron emitters, are... 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

Results from measurements of time-dependent effects depend on the sample history and experimental conditions and should be considered approximate. For example, the state of an unsheared or undisturbed sample is a function of its previous shear history and the length of time since it underwent shear. The area of a thixotropic loop depends on the shear range covered, the rate of shear acceleration, and the length of time at the highest shear rate. However, measurements of time-dependent behavior can be usehil in evaluating and comparing a number of industrial products and in solving flow problems. 

Skaret presents a general air and contaminant mass flow model for a space where the air volume, ventilation, filtration, and contaminant emission have been divided for both the zones and the turbulent mixing (diffusion) between the zones is included. A time-dependent behavior of the concentration in the zones with constant pollutant flow rate is presented. 

The form of Equation (6.7) reveals an interesting aspect of slow binding inhibiton due to enzyme isomerization. A slow forward isomerization rate is insufficient to result in slow binding behavior. The reverse isomerization rate must also be slow, and in fact must be significantly slower that the forward isomerization rate. If this were not the case, there would be no accumulation of the E I conformation at equilibrium. As the value of k6 becomes k5, the denominator of Equation (6.7) approaches unity. Hence the value of Kf approaches Kit and one therefore does not observe any time-dependent behavior. 

In reaction rate studies one s goal is a phenomenological description of a system in terms of a limited number of empirical constants. Such descriptions permit one to predict the time-dependent behavior of similar systems. In these studies the usual procedure is to try to isolate the effects of the different variables and to investigate each independently. For example, one encloses the reacting system in a thermostat in order to maintain it at a constant temperature. 

In order to indicate the types of complications involved in proceeding from this point to equations that indicate the time-dependent behavior of the various species concentrations, it is instructive to consider the following rate expressions. 

Obviously the curve depicting the time-dependent behavior of the concentration of species C can take on even more forms than that for species B. The shape of the curve is dependent on the initial concentrations of the various species and the three reaction rate constants. Figure 5.3 depicts the time-dependent behavior for the specific case where only species A is present initially. 

If the first reaction is regarded as zero-order irreversible (i.e., the enzyme is saturated with substrate), and the second reaction is first-order in the product B, determine the time-dependent behavior of the concentration of species B if no B is present initially. How long does it take to reach 98% of the steady-state value if kx = 0.833 mole/m3-ksec and k2 = 0.767 sec 1 What is this steady-state value ... 

If one were to start with pure linoleic acid, sketch the time dependent behavior of all species in terms of normalized concentrations Ci/CA0). 

Niemantsverdriet, J. W., and van der Kraan, A. M. 1981. On the time-dependent behavior of iron catalysts in Fischer-Tropsch synthesis. J. Catal. 72 385-88. 

This section mainly builds upon classic biochemistry to define the essential building blocks of metabolic networks and to describe their interactions in terms of enzyme-kinetic rate equations. Following the rationale described in the previous section, the construction of a model is the organization of the individual rate equations into a coherent whole the dynamic system that describes the time-dependent behavior of each metabolite. We proceed according to the scheme suggested by Wiechert and Takors [97], namely, (i) to define the elementary units of the system (Section III. A) (ii) to characterize the connectivity and interactions between the units, as given by the stoichiometry and regulatory interactions (Sections in.B and II1.C) and (iii) to express each interaction quantitatively by... 

We seek to describe the time-dependent behavior of a metabolic network that consists of m metabolic reactants (metabolites) interacting via a set of r biochemical reactions or interconversions. Each metabolite S, is characterized by its concentration 5,(f) > 0, usually measured in moles/volume. We distinguish between internal metabolites, whose concentrations are affected by interconversions and may change as a function of time, and external metabolites, whose concentrations are assumed to be constant. The latter are usually omitted from the m-dimensional time-dependent vector of concentrations S(t) and are treated as additional parameters. If multiple compartments are considered, metabolites that occur in more than one compartments are assigned to different subscripts within each compartment. 

The first term in Eq. (68) describes the steady-state properties of the system, as exploited by flux balance analysis to constrain the stoichiometrically feasible flux distributions. Since we consider infinitesimal perturbations only, quadratic terms in the expansion are neglected. In this case, the time-dependent behavior of an infinitesimal perturbation AS(t) = S — S° in the vicinity of S° is described by a linear differential equation... 

The Jacobian matrix of any metabolic network can be written as product of two matrices [23, 84, 325]. Consider the metabolic balance equation, describing the time-dependent behavior of the concentration. S, -(7),... 

The time-dependent behaviors of metal ions adsorption were measured by varying the equilibrium time between the adsorbate and adsorbent in the range of 30-300 min. The concentration of Pb(ll) and Zn(ll) were kept as 50 ttg/mL while the amount of resin added was 0.5 g. The experiments were performed at pH 4 for Pb L... 

Various adsorption parameters for the effective removal of Pb + and ions by using new synthesized resin as an adsorbent from aqueous solutions were studied and optimized. Time-dependent behavior of Pb + and ions adsorption was measured by varying the equilibrium time between in the range of 30-300 min. The percentage adsorption of Pb + plotted in Fig. 26.2 as a function of contact time... 

The time-dependent behavior of ions adsorption was measured by varying the equilibrium time between the adsorbent (ground sumac leaves) and adsorbent (Cu " ions) in the range of 30 min and 24 h. The concentration of was kept 40 pg ttiL , particle size 710 pm, and amount of adsorbent 0.1 g. 

Time-depended behavior of Cu + ion adsorption was measured by varying the equilibrium time between in the range of 0.5-72 h. The percentage adsorption of Cu + ions plotted in Fig. 28.2 as a function of contact time. The percentage adsorption of Cu + indicates that the equilibrium between the Cu + ions and sumac leaves was attained 4 h. Therefore, 4 h stirring time was found to be appropriate for maximum adsorption and was used in all subsequent measurement. The effect of temperature and pH the adsorption equilibrium of Cu + on sumac leaves was investigated by varying the solution temperature from 283 to 303 and pH from 6 to 10. The results are presented in Fig. 28.3. The results indicated that the best adsorption results were obtained at pH 8 at 293 K. 

Probably the best way to illustrate what we mean by process dynamics and control is to take a few real examples. The first example describes a simple process where dynamic response, the time-dependent behavior, is important. The second example illustrates the use of a single feedback controller. The third example discusses a simple but reasonably typical chemical engineering plant and its conventional control system involving several controllers. 

Dynamics. Time-dependent behavior of a process. The behavior with no controllers in the system is called the openloop response. The dynamic behavior with feedback controllers included with the process is called the closedloop response. 

I a this section we will study the time-dependent behavior of some chemical. engineering systems, both openloop (without control) and closedloop (with controllers included). Systems will be described by diflerential equations, and solutions will be in terms of time-dependent functions. Thus, our language for this part of the book will be English. In the next part we will learn a little Russian in order to work in the Laplace domain where the notation is more simple than in English. Then in Part V we will study some Chinese because of its ability to easily handle much more complex systems. 

The dynamic relationships discussed thus far in this book were determined from mathematical models of the process. Mathematical equations, based on fundamental physical and chemical laws, were developed to describe the time-dependent behavior of the system. We assumed that the values of all parameters, such as holdups, reaction rates, heat transfer coeflicients, etc., were known. Thus the dynamic behavior was predicted on essentially a theoretical basis. 

The second category comprises the flash photolysis experiments using the short high power light pulses from Q-switched lasers, furthermore all investigations of time-dependent behavior of excited dye molecules, which play an important role as active material in dye lasers or as saturable absorbers in passive Q-switched giant pulse lasers. 

The time and temperature dependent properties of crosslinked polymers including epoxy resins (1-3) and rubber networks (4-7) have been studied in the past. Crosslinking has a strong effect on the glass transition temperature (Tg), on viscoelastic response, and on plastic deformation. Although experimental observations and empirical expressions have been made and proposed, respectively, progress has been slow in understanding the nonequilibrium mechanisms responsible for the time dependent behavior. 

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