Bubble point equation

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 ... 

This is the bubble-point equation for a three-component system. 

Are we missing the pressure PT in the flash drum shown in Fig. 9.2 Let s calculate this pressure, using the bubble-point equation and the vapor pressure chart shown in Fig. 9.1 ... 

At pressures below the bubble point, Equation 8-24 applies. 

The coefficient of isothermal compressibility of a liquid is defined in Chapter 8. Equations 8-7 apply to a liquid at pressures above its bubble point. Equation 8—24 applies to a reservoir liquid at pressures below its bubble point. Figure 8-7 shows the effect a decline in reservoir pressure has on oil compressibility. 

The bubble- and dew-point equations. The equilibrium equation (Eq. 4.3) and the composition constraint [Eq. (4.1)] are combined to get the bubble-point equation,... 

The BP methods use a form of the equilibrium equation and summation equation to calculate the stage temperatures, The first BP method, by Wang and Henke (24), included the first presentation of the tridiagonal method to calculate the component flow rates or compositions. These are used to calculate the temperatures by solving the bubble-point equation but this temperature calculation can be prone to failure. 

An alternative is to use the tridiagonal method for calculating compositions, but to calculate the new temperatures directly, without iterating on the bubble-point equation, These new temperatures are approximate but as long as the internal compositions are properly corrected during each column trial, the temperature profile will continue to move toward the solution. This is the basis of the theta method of Holland (7, 9, 26). With either alternative, the energy balances are used to find the total flow rates. 

The theta method. This method has been primarily applied to the Thiele-Geddes equations but a form of the theta method equation has also been applied to the equations of the Lewis-Matheson method. The main independent variable of the method is a convergence promoter, theta (or 6). The convergence promoter 0 is used to force an overall component and total material balance and to adjust the compositions on each stage. These new compositions are then used to calculate new stage temperatures by an approximation of the dew- or bubble-point equation called the Kb method. The power of the Kb method is that it directly calculates a new temperature without the sort of failures that occur when iteratively solving the bubble- or dew-point equations. 

The Kb method. For updating the tray temperatures, the theta method relies on the Kb method. The Kb method takes advantage of the near-linear dependence of the logarithm of the K-values and the relative volatilities on temperature over short temperature spans. Relative volatilities (a s) are calculated with respect to a base component K-value, K.bj k, at the stage temperature of the current column trial, Tjk. The base component is usually a middle boiler or a hypothetical component, The K-value of the base component for the next trial, Kbjk + 1( is calculated using a form of the bubble-point equation unique to the Kb method ... 

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). 

When the vapor phase is ideal, the Kt are independent of the vapor composition. In such a case, the procedure for bubble-point determination is to (1) guess a temperature (2) calculate the K, which equal Yiff/P, where yt is the activity coefficient of the ith component in the liquid phase, f is the fugacity of pure liquid i at system temperature and pressure, and P is the system pressure and (3) check if the preceding bubble-point equation is satisfied. If it is not, repeat the procedure with a different guess. 

In the Newton-Raphson methods, the bubble point equations and energy balances are solved simultaneously for the stage temperatures and vapor flow rates the liquid flow rates follow from the total material balances [Tierney and coworkers, AlChE J., 13, 556 (1967) 15, 897 (1969) Billingsley and Boynton, AIChE /., 17, 65 (1971)]. Similar methods are described by Holland (op. cit.) and by Tomich [AIChE J., 16,229 (1970)]. 

Microporous membranes will fill their pores with wetting fluids by imbibing that fluid in accordance with the laws of capillary rise. The retained fluid can be forced from the filter pores by air pressure applied from the upstream side. The pressure is increased gradually in increments. At a certain pressure level, liquid will be forced first from the set of largest pores, in keeping with the inverse relationship of the applied air pressure P and the diameter of the pore, d, described in the bubble point equation ... 

The bubble point equation is essentially the capillary rise equation ... 

The necessary bubble-point equation is obtained in the manner described in Chapter 7 by combining (15-2) and (15-3) to eliminate y,j giving... 

Bubble point equation for multiple components under Raoult s Law... 

We, therefore, assume slow boiling here, and thus at any given liquid composition x, the vapor in equilibrium with it may be solved through the bubble point equation ... 

Application. Both the Tomich and the 2iV Newton-Raphson methods are proven methods and have been applied often. The Tomich method was part of the GMB system of The Badger Company, Cambric, Massachusetts, and is in many in-house simulators. Both methods are best for wide- or middle-boiling s iarations. Because one of the equations in the 2N Newton-R hson method is a dew- or bubble-point equation, it may work better for middle or more narrow-boiling mixtures than the Tomich method. Both methods have also been eq>plied to absorber-strippers, thou an SR method is still the best method for the most wide-boiling absorber-strippers. Because of the full Jacobian more numbers to manipulate), for columns over 50 stages these methods will use excessive computer time and memoiy. Also, the solution of the Jacobian is prone to failure when the number of stages is high, and so these methods are not recommended for tall columns. 

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