PI approach

The chemical potential pi plays a vital role in both phase and chemical-reaction equilibria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence pi, is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, pi approaches negative infinity when either P or Xi approaches zero. While these characteristics do not preclude the use of chemical potentials, the application of equilibrium criteria is facilitated by introduction of the fugacity, a quantity that takes the place of p. but which does not exhibit its less desirable characteristics. 

When the reduced pressure and temperature approach 1.0, P,/Pi approaches the limiting value of 0.606. 

The ratio of the fugacity/2 at the pressure P2 to the fugacity/i at the pressure Pj can be obtained by graphical or numerical integration, as indicated by the area between the two vertical lines under the isotherm for the real gas in Figure 10.6. However, as Pi approaches zero, the area becomes infinite. Hence, this direct method is not suitable for determining absolute values of the fugacity of a real gas. 

As the dimensionless concentration of the reactant decreases so that pi just passes through the upper Hopf bifurcation point pi in Fig. 3.8, so a stable limit cycle appears in the phase plane to surround what is now an unstable stationary state. Exactly at the bifurcation point, the limit cycle has zero size. The corresponding oscillations have zero amplitude but are born with a finite period. The limit cycle and the amplitude grow smoothly as pi is decreased. Just below the bifurcation, the oscillations are essentially sinusoidal. The amplitude continues to increase, as does the period, as pi decreases further, but eventually attains a maximum somewhere within the range pi% < pi < pi. As pi approaches the lower bifurcation point /zf from above, the oscillations decrease in size and period. The amplitude falls to zero at this lower bifurcation point, but the period remains non-zero. 

The standard state chosen in the foregoing is a hypothetical one, viz, an actual gas behaving ideally at 1 atm. pressure, which may be difficult to comprehend. Its use may be avoided, however, by means of a reference state which leads to exactly the same results. The reference state chosen is identical with that employed in connection with fugacity, namely, the gas at very low total pressure of the mixture. It is then postulated that the ratio of the ojctivity of any gas to its partial pressure becomes unity in the refer--ence state, i.e., apd)/pi approaches unity as the total pressure becomes very small. It will be evident that this postulate makes the activity of a gas in a mixture identical with its fugacity, just as does the standard state proposed above. 

Figure 3.4 Relevant issues in the main four domains that constitute the pillars of the PI approach. Source adapted from EU Roadmap for Process Intensification ...

Parallel to PI, a number of equipment vendors have been developing a number of interesting new apparatuses. This new equipment is mainly focussed on the reaction section in the chemical process. Although the size of plant often is determined by the downstream processing part, i.e. separating sections such as distillation and extraction, a first step in the PI approach might be the size reduction of the reactor. 

For large values of parameter Tx/L, when currents practically vanish in the formation, curves of Pr/pi approach to their right-hand asymptotes, equal to P2/P1, if P2 + 00. 

Severe diffusion limitation at large values of q>i (for q>i > 3), where tanh surface reaction is fast (large fcv)> and/or the diffusion is slow (small Dcomb)... 

The principal suppliers of intensified unit operations increasingly realise that the ideal approach is driven by business requirements, although of course process drivers will retain an important place. Process drivers are those where the physical and chemical requirements of the process are determined and then used to select the equipment which best suits the process. Business drivers are financial issues relating to the operability and profitability of the process, and companies such as Protensive rightly see the massive business opportunities afforded by the radical PI approach. 

This sort of quantity will be found throughout this chapter denoted by making reference to the number X of significant beads utilized in the corresponding PI approach studied, which will be X = P, or X = P / 2 (P even). 

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