Linear mixing law

Differential volatility is not the only physics responsible for the foot feature of binary solvents. Flows due to surface tension gradients are clearly present. A simple linear mixing law (known to underestimate the gradients potentially present) for the total surface tension a in terms of its constituents serves to estimate the magnitude of the flow,... 

Linear mixing law is valid in volume proportional quantities. And this also applies to volume specific heat capacity,... 

The sidestream column can be broken into a series of CSs in an analogous manner to Figure 6.1. Since sidestream withdrawal is essentially the reverse of multiple feed additions, the difference points obey the same linear mixing laws as shown in Figure 6.19. Therefore, all difference points and compositions are linked to each other via straight lines in composition space. 

The approach to constructing adiabatic ARs with temperature is slightly more complicated than isothermal constructions. The energy balance generally does not allow for temperature to obey a linear mixing law, and as a result temperature cannot generally be treated as a pseudo component in the state vector C (which is possible with residence time t). If temperature is to be incorporated, T must usually be introduced into the analysis via an energy balance and treated as an extra parameter in a rate expression, in the form r(C, T). 

A negative value indicates a net release of energy. Therefore, 140.3 kJ of cooling is required per kg of mixture in order for temperature to obey a linear mixing law. 

It is generally not possible to mix temperature in the same fashion as with concentration or residence time. This is because temperature generally does not obey a linear mixing law (unless the energy balance is simple). 

The form of Equation 5.10 is identical to the equation describing the linear mixing law for concentrations developed in Chapter 2. The process of combining two parallel reactors (or reactor networks) of different residence time results in a linear mixing law as well. This implies that residence time may be used in the construction of candidate ARs in a similar manner to that for concentration. 

Since r also obeys a linear mixing law, residence time may be incorporated into AR constructions and treated as if it is a component in the concentration vector C. That is, T may be viewed as a pseudo component in C. 

Figure 5.21 Geometric interpretation of mixing in residence time space. Mixtures also lie on a straight line when residence time is used since residence time also obeys a linear mixing law.

Points 1 and 2 are associated with the state vectors Cj = [Cj, Tj] and C2 = [C2, T2F, respectively. The mixture is hence given by the state vector C = [c, r ]. Geometrically, C must lie on the straight line joining states Cl and C2 in c-r space as both concentration and residence time obey a linear mixing law. 

Constructions involving minimum residence time (minimum reactor volume) were also discussed. These constructions are effectively identical to those constructed in concentration space (the phase plane). This is feasible as residence time behaves as a pseudo component in concentration space—r obeys a linear mixing law and it is easily incorporated into the rate vector r(C). 

For this example, a relation to describe how temperature changes with mixing is required. For simplicity, you may assume that temperature obeys a linear mixing law, so that T = ATi-I-(1-A)T2. 

Since is it assumed that temperature obeys a linear mixing law, we can use the values of A provided to compute mixture concentrations as well as temperatures that act as feed conditions to the second reactor. Therefore,... 

For example, a triangle, representing 8 in two-dimensional concentration space, might be viewed as a triangular prism in C-t space as in Figure 8.6. A similar argument applies to temperature, when temperature can be assumed to obey a linear mixing law. 

All of the systems and associated AR constructions provided thus far have been computed, primarily, in concentration space. Understanding concepts related to concentration is more intuitive, we believe. Thus, it is simpler to base discussions in concentration space, as opposed to other process variables that also obey linear mixing laws. The AR has also been developed historically with isothermal constant density systems in mind, although this constraint is relaxed in Chapter 7. Yet, concepts such as those from Chapters 6 to 8 demonstrate that even when these constraints are relaxed, the resulting theory might still be quite complex. 

Defining a species mole fraction x,- in the feed may contain different relative proportions of component i when compared to the product stream. The mole fraction X, in the feed is hence not directly comparable to x, in the product stream. If we were to mix product and feed streams, the resulting mixture may not obey a linear mixing law as a consequence. 

To show that residence time in mass fraction space obeys a linear mixing law, the traditional definition for t written in terms of volumetric flow rate is no longer suitable. Instead, we define, in an analogous fashion, the equivalent mass fraction residence time for a reactor i as follows ... 

A number of simple transformations were explained in this chapter that help to express common quantities used in AR theory in terms of species mass fractions. Mass fractions in AR computations are useful for describing industrial systems, as the conservation of mass guarantees that mixing in mass fraction space always obeys a linear mixing law. This result, in turn, allows for the use of AR theory in variable density systems (when molar concentration does not mix linearly). 

Chapter 9 discussed the construction of candidate ARs in variable density systems. Mass fractions are hence useful in broadening the use of AR theory to a wider class of systems (i.e., gas phase reactions). Mass fractions always obey a linear mixing law, and thus these constructions are valid even when the system does not obey constant density. We described a number of useful relations for converting common process variables used in AR theory, such as concentration and mole fraction, in terms of species mass fractions. These conversions are often assisted by use of an appropriate equation of state that relates the system volume to the process variables of interest. Rate expressions involving concentration are then easily viewed in mass fraction space, and the AR may be computed appropriately. 

Saalwachter, K., Multiple-Quantum Nuclear Magnetic Resonance Investigations of Molecular Order in Polymer Networks Evidence for a Linear Mixing Law in Bimodal Systems. J. Chem. Phys. 2003,119,3468-3482. 

K. Saalwachter, P. Ziegler, O. Spyckerelle, B. Haidar, A. Vidal, J.-U. Sommer, IH multiple-quantum nuclear magnetic resonance investigations of molecular order distributions in poly(dimethylsiloxane) networks evidence for a linear mixing law in bimodal systems, J. Chem. Phys. 119 (2003) 3468—3482. 

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