Bubble-point equation temperature calculation using

The bubble point equation can be used to back calculate/estimate interfacial temperature. Using the effective pore diameter for a 325 x 2300 screen from Chapter 4, the methane surface tension curve fitting parameters, and the experimental bubble point data using the three different pressurant gases from the current chapter, one can estimate the interfacial temperature ... 

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). 

Why not put new lyrics to an old tune This is an excellent idea, and many have done this very thing. Rice" started w ith the Smith-Brinkley raethod" used to calculate distillation, absorption, extraction, etc., overhead and bottoms compositions, and developed distillation equations for determining the liquid composition on any tray. This together with bubble point calculations yield a column temperature profile useful for column analysis. 

If the K-value requires the composition of both phases to be known, then this introduces additional complications into the calculations. For example, suppose a bubble-point calculation is to be performed on a liquid of known composition using an equation of state for the vapor-liquid equilibrium. To start the calculation, a temperature is assumed. Then, calculation of K-values requires knowledge of the vapor composition to calculate the vapor-phase fugacity coefficient, and that of the liquid composition to calculate the liquid-phase fugacity coefficient. While the liquid composition is known, the vapor composition is unknown and an initial estimate is required for the calculation to proceed. Once the K-value has been estimated from an initial estimate of the vapor composition, the composition of the vapor can be reestimated, and so on. 

The system methanol-cyclohexane can be modeled using the NRTL equation. Vapor pressure coefficients for the Antoine equation for pressure in bar and temperature in Kelvin are given in Table 4.176. Data for the NRTL equation at 1 atm are given in Table 4.186. Assume the gas constant R = 8.3145 kIkmol 1-K 1. Set up a spreadsheet to calculate the bubble point of liquid mixtures and plot the x-y diagram. 

Again we will consider that the quantity of gas at the bubble point is negligible. Thus we can substitute ng = 0 and nL = n into Equation 12-16 to obtain an equation which can be used to calculate the bubble-point pressure at a given temperature or the bubble-point temperature at a given pressure. 

Phase envelopes for typical natural gas tend to be fairly broad. That is, they cover a large range of temperature and pressure. On the other hand, the phase envelopes for acid gas mixtures tend to be quite narrow. Figure 3.2 shows the phase envelope for a mixture containing 50 mol% H2S and 50 mol% C02. This phase envelope was calculated using the Peng-Robinson equation of state, and the bubble, dew, and critical points are labeled. 

Calculate the bubble-point temperature. Assume a temperature and then calculate values for the equilibrium relations fi om Equations 3.3.22 and 3.3.23 in Table 3.2.2. Next, calculate the vapor-phase mole fractions fi om Equations 3.3.20 and 3.3.21. Check the results using Equation 3.3.19. Assume a new ten ierature and repeat the calculation until temperature converges to a desired degree of accuracy. 

To obtain the composition of the top and bottom products, first calculate the relative volatility of each component using the conditions of the feed as a first guess. The relative volatility depends on temperature and pressure. The bubble point of the feed at 400 psia (27.6 bar) and at the feed composition, calculated using ASPEN [57], is 86.5 °F (130 °C). The K-values of the feed are listed in Table 6.7.1. Bubble and dew points could also be calculated using K-values from the DePriester charts [31] and by using the calculation procedures given in Chapter 3. Next, calculate the relative volatility of the feed stream, defined by Equation 6.27.18, for each component relative to the heavy key component. 

After calculating the temperature of the top and bottom products, obtain a new estimate of the colmnn relative volatility for each component. Find the relative volatihty of each conponent in the bottom and top product. Assuming that we have a total condenser, the conposition of the vapor rising above the top tray is equal to the conposition of the top product. The calculation for the dew-point tenperature will give the composition of the hquid on the top tray as well as the temperature. The temperature and hquid composition at the bottom tray is obtained from a bubble point calculation. Next, calculate the relative volatility of each conponent at the top and bottom tray. Using these values of the relative volatihty and the values for the feed, calculate the geometric average volatihty, (oCj)avg, of each component from Equation 6.26.19. This calculation is summarized in Table 6.7.2... 

As simple as these equations are, they result in some unexpected problems. Consider first the case where is assigned. The practical problem becomes the determination of the bubble point at some given temperature. The pressure corresponding to the bubble point is —that part is easy. One can now renormalize to x = bIp, and use Eq. (33) to calculate the vapor phase composition X (x). Notice, however, that X (a ) will not have the same functional form as X (x) while the latter has a first moment equal to unity, = 1, the former has (- l)th moment equal to unity, = 1. If, for instance, one uses a gamma distribution,. J (x) = a,x), (x) will not be given by a gamma distribution. 

Ahernate Method for Catculaling the Bubble-Point Pressure of an Ideal Two-Component System. Although Raoult s Law can be used directly to calculate the bubble-point pressure of an ideal solution, an alteiTiate method which is applicable to two-component systems will now be presented. Since equations 5 to 8 ai e applicable an3rwhere in the two-phase region they apply at the bubble point and the dew point. At the bubble point the system is essentially all liquid except for an infinitesimal amount of vapor. Consequently, the composition of the liquid will be equal to the overall composition of the system. If the overall composition is substituted for x and Xz in equations 5 and 6 then either may be solved for Pt at a given temperature. The value of Pt calculated in this manner is equal to the bubble-point pressure. The com position of tlie infinitesimal amount of vapor at the bubble point may be computed by substitution in equations 7 and 8. 

The calculations were done using vapor pressure data available in Reid et al. (1987) for each of the species Table VI gives the results. The temperature selected for evaluating the vapor pressure is the bubble point at 1 atm for the feed mixture, i.e., at 434.21 K. Using the relative volatilities and feed flows shown in Table VI, we can estimate the marginal vapor rates shown in Table VII using the equation... 

The program VDWMIX is used to calculate multicomponent VLE using the PRSV EOS and the van der Waals one-fluid mixing rules (either IPVDW or 2PVDW see Sections 3.3 to 3.5 and Appendix D.3). The program can be used to create a new input file for a multicomponent liquid mixture and then to calculate the isothermal bubble point pressure and the composition of the coexisting vapor phase for this mixture. In this mode the information needed is the number of components (up to a maximum of ten), the liquid mole fractions, the temperatures at which the calculations are to be done (for as many sets of calculations as the user wishes, up to a maximum of fifty), critical temperatures, pressures (bar), acentric factors, the /f constants of the PRSV equation for each compound in the mixture, and, if available, the experimental bubble point pressure and the vapor phase compositions (these last entries are optional and are used for a comparison between the experimental and calculated results). In addition, the user is requested to supply binary interaction parameteifs) for each pair of components in the multicomponent mixture. These interaction parameters can be... 

Equation 7-29 is used to determine the bubble point temperature and pressure. In using Equation 7-29, the temperature or pressure is fixed, while the other parameter is varied until the criterion for a stable system is satisfied. A combination of temperature and pressure is altered, if the summation of the calculated vapor composition is different from unity. There is no known direct method that will allow a reasonable estimation of the amount of change required. However, Dodge [4], Hines and Maddox [5] have provided techniques for reducing the number of trials that is required. 

When the distribution coefficients are composition-dependent, the above method must be modified to account for the effect of composition. A search for the unknown bubble point or dew point temperature or pressure is started on the basis of some composition-independent relationship between the X-values and the temperature and pressure, such as Equations 2.20 and 2.21. Component fugacities are then calculated for the vapor phase and the liquid phase, and the /f-values are updated using Equation 2.15. The calculations are repeated until Equation 2.16 or 2.17, as well as Equation 2.12, are satisfied. The iterative scheme for the bubble point pressure calculation may proceed along the following steps ... 

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