Bubble-point equation deriving

The purpose of this chapter is to explain what is meant by the terms bubble point and dew point, and how we can use these ideas to improve the operation of the distillation tower. To begin, we will derive the bubble-point equation, from the basic statement of vapor-liquid equilibrium ... 

Christiansen et al. (54) applied the Naphtali-Sandholm method to natural gas mixtures. They replaced the equilibrium relationships and component vapor rates with the bubble-point equation and total liquid rate to get practically half the number of functions and variables [to iV(C + 2)]. By exclusively using the Soave-Redlich-Kwong equation of state, they were able to use analytical derivatives of revalues and enthalpies with respect to composition and temperature. To improve stability in the calculation, they limited the changes in the independent variables between trials to where each change did not exceed a preset maximum. There is a Naphtali-Sandholm method in the FraChem program of OLI Systems, Florham Park, New Jersey CHEMCAD of Coade Inc, of Houston, Texas PRO/II of Simulation Sciences of Fullerton, California and Distil-R of TECS Software, Houston, Texas. Variations of the Naphtali-Sandholm method are used in other methods such as the homotopy methods (Sec. 4,2.12) and the nonequilibrium methods (Sec. 4.2.13). 

The huhble point model has been derived in a way to make it facile for a system designer to predict performance for any screen in any fluid at any thermodynamic state. From Chapter 10, the updated cryogenic bubble point equation is expressed as ... 

This is consistent with Equation 8-9 since the derivative of Rs with respect to pressure is zero at pressures above the bubble point. 

In the preceding sections the possibility of the strong influence of the fi-ee surface zone near the front stagnant point on the transport stage of microflotation was emphasised. Eq. (10.40) enables us to estimate the critical condition of the appearance of an f.s.f.z. The parameters v and Re relate to the bubble with a completely retarded surface and can be described by equations derived for solid spheres. 

A function (f) based on the bubble points of the light and heavy key components (Tuc and Thk) would also be worth considering if other approaches fail. The bubble points (at normal operating pressure) are derived from the Antoine Equation. For the HK inferential Tis the temperature on the tray selected in the upper section of the column. 

This appendix presents computation of resultant solid/liquid and solid/vapor interfacial tensions from the methanol/water binary mixture bubble point data from Chapter 4. Governing equations are presented for deriving the Langmuir isotherms for the S/L and S/V data. The goodness of fits are also discussed for both cases. 

Returning to Figure 8.16, we now consider the case in which T and P at equilibrium are both specified. This is commonly called an isothermal flash, although a much better name would be a T- and P-specifled flash. In the bubble-point calculations, V 0.00, and in the dew-point calculations L K. 0.00. In T- and P-specified flashes both L and V have nonzero values. This means that we have gained two variables, but we have also gained two equations, Eqs. 8.7 and 8.8. (It might appear that we have two Eq. 8.8s, but one of them is derivable from Eq. 8.7 and the other from Eq. 8.8, so only one of the two, Eq. 8.9 below, is independent and useful.)... 

This equation may also be used to calculate the wall thickness distribution in deep truncated cone shapes but note that its derivation is only valid up to the point when the spherical bubble touches the centre of the base. Thereafter the analysis involves a volume balance with freezing-off on the base and sides of the cone. 

In a similar manner, thickness distributions can be derived in other relatively simple but more realistic molds, such as truncated cones. In such cases, the above model holds until the bubble comes in contact with the bottom of the mold at its center. From that point on, new balance equations must be derived. 

Equation (6.42) can be derived in a similar way as Eq. (6.37) assuming no-slip boundary conditions and small distances (D -ti rstarting point for calculating the force between drops and bubbles. 

As discussed above, the fluid equations are derived using the assumption that macroscopic quantities like the solids volume fraction vary slowly in space. In that case, basing closure relations on local macroscopic variables seems reasonable. However, in reality, the solids volume fraction can change quite abrupdy, e.g., at emulsion—bubble interfaces or when particle clusters are formed. It might be essential to incorporate these heterogeneities that arise on mesoscopic length scales into closure relations. We wiU return to this point in Section 4.1. 

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