Bubble-T calculations

From the measurables T, P, x , and jcP, five common phase equilibrium problems can be contrived, depending on which quantities are known and which are unknown. For example, the problem introduced in the previous paragraph involves P and x as known and requires us to solve (11.1.1) for T and xP. When phase a is liquid and phase 3 is vapor, this problem is called a bubble-T calculation, for we are to compute a point on the bubble curve of an isobaric Txy diagram. This along with the other four common problems are listed in Table 11.1. 

In the table the second, third, and fourth problems each result from a permutation of the known and unknown quantities that occur in the bubble-T calculation. We refer to these as P-problems, because each problem is well-posed when values are specified for P independent intensive properties, where the value of T is given by the phase rule (9.1.14). However, the flash problem in Table 11.1 differs from the others in that it is an P -problem it is well-posed when values are specified for T independent intensive properties, with the value of T given by (9.1.12). Flash calculations pertain to separations by flash distillation in which a known amount N of one-phase fluid, having known composition z, is fed to a flash chamber. When T and P of the chamber are properly set, the feed partially flashes, producing a vapor phase of composition xP in equilibrium with a liquid of composition x ). The problem is to determine these compositions, as well as the fraction of feed that flashes NP/N. Unlike the other problems in Table 11.1, the flash problem involves the relative amounts in the phases and therefore a solution procedure must invoke not only the equilibrium conditions (11.1.1) but also material balances. 

To illustrate an implementation of the phi-phi method ( 10.1.1), let us consider the bubble-T calculation for vapor-liquid equilibria. Recall the phi-phi method uses FFF 1 for both liquid and vapor phases, and so, as a prerequisite to the calculation, we must choose an equation of state that reliably correlates the volumetric behavior of both phases. Typical candidates include the Peng-Robinson [7] and Redlich-Kwong-Soave [8] equations. For VLB calculations, the phi-phi equations (10.1.3) can be posed in terms of K-factors,... 

For the bubble-T calculation in the phi-phi form, a viable alternative to Newton-Raphson is presented in Figure 11.1. This algorithm is composed of three principal parts an initialization, an outer loop that searches for the unknown T, and an inner loop that searches for the vapor-phase mole fractions y. The algorithm can be used for any number of components, but it is restricted to equilibrium between two phases. In the special case of a single component, the algorithm is equivalent to the Maxwell equal-area construction given in (8.2.22). 

To illustrate the gamma-phi method ( 10.1.2), we reconsider the bubble-T calculation. Now we intend to use FFF 1 for the vapor phase and one of FFF 2-5 for the liquid. In most cases, we use FFF 5 for the liquid-phase fugacity, then the gamma-phi equations (10.1.4) take the form... 

Application of these equations gives the results in Table 13-12. A set of T is calculated from the normahzed by bubble-point calculations. Corresponding values of are obtained from y = K x. Once newA. andT are available, new values of Vn are calculated from energy balances by using data from Maxwell (Data Book on Hydiocaihons, Van Nostrand, Princeton, N.J., 1950). First, an estimate of condenser duty is computed from an energy balance around the condenser. 

As discussed by Franks (1972), in order to solve this system of equations, a value of temperature T must be found to satisfy the condition that the difference term 6 = P - Zpj is very small, i.e., that the equilibrium condition is satisfied. This is known as a bubble point calculation. The above system of defining equations, however represent, an implicit algebraic loop and the trial and error solution procedure can be very time consuming, especially when incorporated into a dynamic simulation program. 

Many variations are possible around the basic flash calculation. Pressure and V/F can be specified and T calculated, and so on. Details can be found in King7. However, two special cases are of particular interest. If it is necessary to calculate the bubble point, then V/F = 0 in Equation 4.55, which simplifies to ... 

Solution To determine the location of the azeotrope for a specified pressure, the liquid composition has to be varied and a bubble-point calculation performed at each liquid composition until a composition is identified, whereby X = y,-. Alternatively, the vapor composition could be varied and a dew-point calculation performed at each vapor composition. Either way, this requires iteration. Figure 4.5 shows the x—y diagram for the 2-propanol-water system. This was obtained by carrying out a bubble-point calculation at different values of the liquid composition. The point where the x—y plot crosses the diagonal line gives the azeotropic composition. A more direct search for the azeotropic composition can be carried out for such a binary system in a spreadsheet by varying T and x simultaneously and by solving the objective function (see Section 3.9) ... 

BUBL P. The iteration scheme for this simple and direct bubble-point calculation is shown in Fig. 12.12. With reference to a computer program for carrying it out, one reads and stores the given values of T and xt, along with all constants required in evaluation of the Pf yk, and d>k. Since y is not given, we cannot... 

BUBL T. Figure 12.14 shows the iterative scheme for this bubble-point calculation. The given values of P and along with appropriate constants are read and stored. In the absence of T and the yk values, all d>k are se,t equal to unity. Iteration is controlled by T, and for an initial estimate we set... 

Interaction parameters, a(x,t), calculated from the bubble population balance itself, e.g., the total bubble density in flowing and stationary foam, the higher moments of the bubble number density distribution, etc. 

We can predict azeotropic behavior as follows from infinite-dilution /T-values. Using a flowsheeting system, we perform a bubble-point calculation for each species in the mixture. Assuming a mixture contains the species A, B, C, and D, we wish to compute the infinite-dilution L-values for three of the species in the remaining one. For example, we perform a flash calculation where A is dominant and B, C, and D are in trace amounts, using something like a feed composition of 0.99999, 0.000003333, 0.000003333, 0.000003334. It does not... 

Example 3 Detv and Bubble Point Calculations As indicated by Example 2a, a binary system in vapor/liquid equilibrium has 2 degrees of freedom. Thus of the four phase rule variables T, P, x, and t/i, two must be fixed to allow calculation of the other two, regardless of the formulation of the equilibrium equations. Modified Raoults law [Eq. (4-307)] may therefore be applied to the calculation of any pair of phase rule variables, given the other two. 

A conventional bubble point calculation involves the specification of the liquid mole fractions and pressure the subsequent computation of the vapor-phase mole fractions and the system temperature. For a binary system (and only for a binary system) we may specify the temperature and pressure and compute the mole fractions of both phases. Thus, our first step is to estimate the interface temperature T. The second step is to solve the equilibrium equations for the mole fractions on either side of the interface. This step is, in fact, equivalent to reading the composition of both phases from a T-x-y equilibrium diagram. 

A bubble-point calculation on (x,)i gives T, from which Hl, can be determined. The vapor rate Vo computed from (12-113) is compared to the assumed value. The method of direct substitution is applied until successive values of Vo are essentially identical. Convergence is rapid to give Vo = 532.4 Ibmole/hr, Li = 802.4 Ibmole/hr, and T, (the exhaust bottom-stage-temperature) = 169.8°F (76.6°C). 

The three sets of (Tj, Sj) are fitted to a quadratic equation for Sj in terms of Tj. The quadratic equation is then employed to predict Tj for Sj = 0, as required by (15-20). The validity of this value of Tj is checked by using it to compute Sj in (15-21). The quadratic fit and Sj check are repeated with the three best sets of (Tj,Sj) until some convergence tolerance is achieved, say Tj" - 0.0001, with T in absolute degrees, where n is the iteration number for the temperature loop in the bubble-point calculation, or one can use S, 0.0001 C, which is preferred. 

Application of these equations gives the results in Table 13-12, A set of is calculated from the normalized by bubble-point calculations. Corresponding values of t/ are obtained from, Once new and are available, new values... 

Carry out hybridization overnight under RNase-free coverslips (avoid trapping air bubbles under the coverslip ) with 20 ng of probe per slide (in 80-pL hybridization buffer see Section 3.1., item 6) at the optimum hybridization temperature (T[ ), calculated for each probe (see Notes 13-15). 

First, we distinguished between an T and an T identification of state. Less information is provided by an IF-spedfication than by an F -specification, but in particular situations one or the other may be more appropriate. For example, in vapor-liquid equilibrium calculations, an IF-specification is sufficient to close a bubble-T problem, but an F -spedfication fails to dose an isothermal flash problem. Furthermore, most reaction-equilibrium problems are not dosed by F-spedfications they require F -specifications. We have also illustrated that in some situations an F-specification may be suffident, but an F -spedfication may lead to a more advantageous problem formulation and solution technique. The prindpal pitfall is to apply an F-specification to a problem that demands an F -spedfication, for then the problem is ill-posed. 

Solution The solution will be easier to follow by first examining the Txy graph in Figure 13-5. Temperature affects mutual solubility and the shape of the liquid-liquid phase boundary. Since we have temperature-dependent activity coefficients, it possible compute the liquid-liquid boundary as a function of temperature. This calculation will be performed up to the bubble temperature. Above the bubble temperature the calculation will be done by solving the bubble T problem. 

Start with a bubble-point calculation on the feed, This converges to T = 60°C at p =... 

H3. Usually, relative volatilities are not constant, and determination of the residue curve requires a bubble-point calculation at each time used to integrate Eq. (8-28) to determine the temperature T and the vapor mole fractions. The bubble-point calculation was illustrated in Exanple 5-3 for light hydrocarbons. The K values for these conpounds can be determined from Eq. (2-161 with the constants tabulated in Table 2-3. Develop a spreadsheet that can be used to determine the residue curves for any three of the following light hydrocarbons i-butane, n-butane, i-pentane, n-pentane, and n-hexane. Note If Euler s method, Eq. (8-291. is used, the tolerance on the sum of the yy values must be quite small (e.g., E -9). Find the residue curve for the following problems. 

The bubble-point calculations, the dew-point calculations and the flash calculations tu-e not carried out in practice for any set of P, T, jr, and JC, . Rather, a subset of conditions which allow certain simplifications are selected and thermodynamic calculations are carried out They are ... 

In an iterative computer-based numerical bubble-point calculation of T, X2 ,. ., when P and Xu, X21, ., are known, the following approach is often adopted. 

In Figure 8.16 we first consider a case in which Lf F,V 0. This is the bubble point calculation, in which we are asking the composition of the vapor that is in equilibrium with a liquid of specified composition. For this condition Eq. 8.8 shows that x, = z, for all / all except for an infinitesimal amount of the material in the feed goes to the liquid. Bubble points need not be thought of as flashes we can think of them as given the composition of the liquid phase and either P or T, find the composition of the equilibrium vapor phase and T or P." But the process-design programs normally include them in the flash module, and they have parallels to the other kinds of flashes in Table 8.5 so we include them here. 

This type of bubble-point calculation has no simple graphical solution on the four types of plots shown in the previous examples because the pressure is unknown. Figure 8.17 shows a T-x diagram for ethanol-water at 1 atm, similar to part d in Figures 8.7, 8.8, 8.9, and 8.12 for a pressure of 1.00 atm. In principle, we could have a separate plot of this type for each possible pressure. (If we wanted the equivalent of Figure 8.17 for some other pressure we could make it up by repeating Example 8.10 (below) at that pressure, for a variety of liquid compositions and plotting the results.) With a set of such plots we would try to find the plot on which the specified T and Xa lie exactly on the liquid composition curve. Then we would read the pressure at which that plot was made, and the at that temperature. In... 

Summarizing bubble-point calculations we see that by hand they are easy if T is specihed, and harder if P is specified. Both are easy in computers. We will see below that dew points are more difficult, because the nonideality is in the phase we are looking for, not in the specified phase. 

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.)... 

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