Isothermal trajectory

These four isothermal trajectories are shown in Figure 5-18, a long with plots of r versus X and plots of l/r versus X. It is seen that the rtiiriimum residence time in such a reactor is one in which the temperature decreases to maintain the lowest l/r as the conversion increases from 0 to 1. 

Figure 5-18 Illustration of isothermal trajectories for an exothermic reversible reaction. At the lowest temperature the rate is low but the eqmlibiium conversion is high, while at the highest temperature the initial rate is high but the equilibrium conversion is low. From the 1/r versus X plot at these temperatures, it is evident that T should decrease as X increases to require a minimum residence time,...

Figure 4.11 shows three possible isothermal trajectories in the CTT diagram. Curing at Tg > Tgoo (trajectory a) leads to complete conversion. However, due to the high values of both the reaction heat and the polymerization rate, it is usually not possible to keep isothermal conditions when curing at high temperatures. The temperature of the sample continuously increases as it is... 

The problem of anomalous behavior of core-softened fluids was widely discussed in literature (see, for example. Ref. [46]). It was shown that for some systems the anomalies take place, while for others do not. In this respect, the question of criteria of anomalous behavior appearance remains the central one. However, another important point is still lacking in the literature—the behavior of anomalies along different thermodynamic trajectories. Here, we call as trajectory a set of points belonging to some path in P, p, T) space. For example, the set of points belonging to the same isotherm we call as isothermal trajectory or briefly isotherm. 

This equation of motion should sample an isothermal trajectory at a temperature determined by the initial internal velocities. Eq. (46) can also be derived directly fromEq. (18) by imposing T(t) = 0. Using Eqs. (8) and (10) this becomes... 

If the coiiplin g parameter (the Bath relaxation constan t in IlyperChem), t, is loo Tight" (<0.1 ps), an isokinetic energy ensemble results rather than an isothermal (microcan on leal) ensemble. The trajectory is then neither canonical or microcan on-ical. You cannot calculate true time-dependent properties or ensemble averages for this trajectory. You can use small values of T for Ih CSC sim ii lalion s ... 

Characteristics of the air jet in the room might be influenced by reverse flows, created by the jet entraining the ambient air. This air jet is called a confined jet. If the temperature of the supplied air is equal to the temperature of the ambient room air, the jet is an isothermal jet. A jet with an initial temperature different from the temperature of the ambient air is called a nonisother-mal jet. The air temperature differential between supplied and ambient room air generates buoyancy forces in the jet, affecting the trajectory of the jet, the location at which the jet attaches and separates from the ceiling/floor, and the throw of the jet. The significance of these effects depends on the relative strength of the thermal buoyancy and inertial forces (characterized by the Archimedes number). 

For rhe non isothermal linear air jet, the trajectory equation is derived bv Shepelev" is... 

Equilibrium Compositions for Single Reactions. We turn now to the problem of calculating the equilibrium composition for a single, homogeneous reaction. The most direct way of estimating equilibrium compositions is by simulating the reaction. Set the desired initial conditions and simulate an isothermal, constant-pressure, batch reaction. If the simulation is accurate, a real reaction could follow the same trajectory of composition versus time to approach equilibrium, but an accurate simulation is unnecessary. The solution can use the method of false transients. The rate equation must have a functional form consistent with the functional form of K,i,ermo> e.g., Equation (7.38). The time scale is unimportant and even the functional forms for the forward and reverse reactions have some latitude, as will be illustrated in the following example. 

We next examine trajectories on these graphs. Two of these are simple straight lines a vertical line at temperature Tc for the isothermal (UA OO) reactor and a straight line... 

Figures—14 Possible region of trajectories for endothermic and for reversible reactions, starting at feed temperature Tj, with heating from the wall at temperature Ti,. Trajectories must be in the shaded region between the adiabatic and isothermal curves and below the equilibrium curve.

Another way of visualizing the optimal trajectory for the exothermic reversible reaction is to consider isothermal reaction rates at increasing temperatures. At T the equilibrium conversion is high but the rate is low, at T2 the rate is higher but the equihbrium conversion is lower, at the rate has increased further and the equihbrium conversion is even lower, and at a high temperature T4 the initial rate is very high but the equihbririm conversion is very low. 

The use of a timescale instead of a conversion one requires a previous definition of the cure schedule e.g., isothermal, constant heating rate, etc. Usually, isothermal conditions are selected to define the timescale i.e., only trajectories at constant temperature have a physical meaning. This leads to the TTT diagram. 

Note that we have said nothing about the size of the reactor vessel. If the reactor operates isothermally, the composition profiles shown in Figures 4.18 and 4.19 are independent of reactor size. Of course, attaining a constant-temperature trajectory becomes more difficult as the vessel size increases because of the reduction in area-to-volume ratio. 

To compare reactions with different time constants it is useful to plot them as trajectories in a multi-dimensional phase space whose coordinates are the species concentrations and the temperature. Fig. 2 shows trajectories projected onto the temperature vs. [r] plane for reactions with identical initial fuel and air concentrations but different initial radical concentrations and temperature. Trajectories beginning at the left had no initial radicals, and the trajectory starting at 1200 K is represented in Fig. 1. The exponential increase of [r] to [r]e is isothermal so it appears horizontal in Fig. 2. The knee of the curve represents the relatively flat portion of Fig. 1 where [r] is approximately [R]e. As the temperature increases [r] remains approximately equal to [R]e, which lies to the left of the dashed line due to consumption of fuel and oxygen. 

Some examples are shown on the Figures 8a-d. All of these trajectories involve the transition from stable to the unstable part of the isotherm. 

The condition u(0 ) = u prescribed by Equation 8, hereafter called the input condition states that every phase trajectory starts on a vertical line u = u. At the same time, the condition v(x ) = vt imposed by Equation 9 which we shall call the output condition demands that every trajectory ends on a horizontal line v = v- . Thus we have a point (u, v ) associated with each solution. The set of all such points constitutes a structure which we call the input - output space of the reactor. For the case of convex isotherm, this was done by Viswanathan and Aris (1). 

---