Calculating results calculations

FIGURE 2.28 Models of a lid with integrated temperature control and calculation results, calculated with Cadmould 3D-F... 

The accuracy of our calculations is strongly dependent on the accuracy of the experimental data used to obtain the necessary parameters. While we cannot make any general quantitative statement about the accuracy of our calculations for multicomponent vapor-liquid equilibria, our experience leads us to believe that the calculated results for ternary or quarternary mixtures have an accuracy only slightly less than that of the binary data upon which the calculations are based. For multicomponent liquid-liquid equilibria, the accuracy of prediction is dependent not only upon the accuracy of the binary data, but also on the method used to obtain binary parameters. While there are always exceptions, in typical cases the technique used for binary-data reduction is of some, but not major, importance for vapor-liquid equilibria. However, for liquid-liquid equilibria, the method of data reduction plays a crucial role, as discussed in Chapters 4 and 6. 

At pressures to a few bars, the vapor phase is at a relatively low density, i.e., on the average, the molecules interact with one another less strongly than do the molecules in the much denser liquid phase. It is therefore a common simplification to assume that all the nonideality in vapor-liquid systems exist in the liquid phase and that the vapor phase can be treated as an ideal gas. This leads to the simple result that the fugacity of component i is given by its partial pressure, i.e. the product of y, the mole fraction of i in the vapor, and P, the total pressure. A somewhat less restrictive simplification is the Lewis fugacity rule which sets the fugacity of i in the vapor mixture proportional to its mole fraction in the vapor phase the constant of proportionality is the fugacity of pure i vapor at the temperature and pressure of the mixture. These simplifications are attractive because they make the calculation of vapor-liquid equilibria much easier the K factors = i i ... 

To illustrate calculations for a binary system containing a supercritical, condensable component. Figure 12 shows isobaric equilibria for ethane-n-heptane. Using the virial equation for vapor-phase fugacity coefficients, and the UNIQUAC equation for liquid-phase activity coefficients, calculated results give an excellent representation of the data of Kay (1938). In this case,the total pressure is not large and therefore, the mixture is at all times remote from critical conditions. For this binary system, the particular method of calculation used here would not be successful at appreciably higher pressures. 

Figure 13 presents results for a binary where one of the components is a supercritical, noncondensable component. Vapor-phase fugacity coefficients were calculated with the virial... 

Table 3 shows results obtained from a five-component, isothermal flash calculation. In this system there are two condensable components (acetone and benzene) and three noncondensable components (hydrogen, carbon monoxide, and methane). Henry s constants for each of the noncondensables were obtained from Equations (18-22) the simplifying assumption for dilute solutions [Equation (17)] was also used for each of the noncondensables. Activity coefficients for both condensable components were calculated with the UNIQUAC equation. For that calculation, all liquid-phase composition variables are on a solute-free basis the only required binary parameters are those for the acetone-benzene system. While no experimental data are available for comparison, the calculated results are probably reliable because all simplifying assumptions are reasonable the... 

Using the method outlined above, calculations were performed for ten ternary systems. All binary parameters are shown in Table 4. Some typical results are shown in Figures 16 to 19. 

Figure 17 shows results for the acetonitrile-n-heptane-benzene system. Here, however, the two-phase region is somewhat smaller ternary equilibrium calculations using binary data alone considerably overestimate the two-phase region. Upon including a single ternary tie line, satisfactory ternary representation is obtained. Unfortunately, there is some loss of accuracy in the representation of the binary VLB (particularly for the acetonitrile-benzene system where the shift of the aceotrope is evident) but the loss is not severe. 

Guffey and Wehe (1972) used excess Gibbs energy equations proposed by Renon (1968a, 1968b) and Blac)c (1959) to calculate multicomponent LLE. They concluded that prediction of ternary data from binary data is not reliable, but that quarternary LLE can be predicted from accurate ternary representations. Here, we carry these results a step further we outline a systematic procedure for determining binary parameters which are suitable for multicomponent LLE. 

Figure 3 presents results for acetic acid(1)-water(2) at 1 atm. In this case deviations from ideality are important for the vapor phase as well as the liquid phase. For the vapor phase, calculations are based on the chemical theory of vapor-phase imperfections, as discussed in Chapter 3. Calculated results are in good agreement with similar calculations reported by Lemlich et al. (1957). ... 

There is a significant difference between the results shown in Figure 2 and calculated results given in Brit. Chem. Eng. Proc. Tech. 16 1036 (1971). We believe the latter to be in error. 

The sum of the squared differences between calculated and measures pressures is minimized as a function of model parameters. This method, often called Barker s method (Barker, 1953), ignores information contained in vapor-phase mole fraction measurements such information is normally only used for consistency tests, as discussed by Van Ness et al. (1973). Nevertheless, when high-quality experimental data are available. Barker s method often gives excellent results (Abbott and Van Ness, 1975). 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

Convergence is usually accomplished in 2 to 4 iterations. For example, an average of 2.6 iterations was required for 9 bubble-point-temperature calculations over the complete composition range for the azeotropic system ehtanol-ethyl acetate. Standard initial estimates were used. Figure 1 shows results for the incipient vapor-phase compositions together with the experimental data of Murti and van Winkle (1958). For this case, calculated bubble-point temperatures were never more than 0.4 K from observed values. 

Appendix C-6 gives parameters for all the condensable binary systems we have here investigated literature references are also given for experimental data. Parameters given are for each set of data analyzed they often reflect in temperature (or pressure) range, number of data points, and experimental accuracy. Best calculated results are usually obtained when the parameters are obtained from experimental data at conditions of temperature, pressure, and composition close to those where the calculations are performed. However, sometimes, if the experimental data at these conditions are of low quality, better calculated results may be obtained with parameters obtained from good experimental data measured at other conditions. 

Now scan a range of values of Ar ,in and calculate the targets for energy, number of units, and network area and combine these into a total cost. The results are given in Table 7.4. 

Another test is the total oxygen demand (TOD) test, which oxidizes the waste in the presence of a catalyst at 900°C in a stream of air. Under these harsh conditions, all the carbon is oxidized to CO2. The oxygen demand is calculated from the difference in oxygen content of the air before and after oxidation. The resulting value of TOD... 

The algorithm may calculate an increase in Qnmin and Qcmin- This means that the match is transferring heat across the pinch or that there is some feature of the design that will cause cross-pinch heat transfer if the design was completed. If the match is not transferring heat across the pinch directly, then the increase in utility will result from the match being too big as a result of the tick-off heuristic. 

Sample size is 100 ml and distillation conditions are specified according to the type of sample. Temperature and volume of condensate are taken simultaneously and the test results are calculated and reported as boiling temperature as a function of the volume recovered as shown in Table 2.1. 

Using this concept, Burdett developed a method in 1955 to obtain the concentrations in mono-, di- and polynuclear aromatics in gas oils from the absorbances measured at 197, 220 and 260 nm, with the condition that sulfur content be less than 1%. Knowledge of the average molecular weight enables the calculation of weight per cent from mole per cent. As with all methods based on statistical sampling from a population, this method is applicable only in the region used in the study extrapolation is not advised and usually leads to erroneous results. 

From the analytical results, it is possible to generate a model of the mixture consisting of an number of constituents that are either pure components or petroleum fractions, according to the schematic in Figure 4.1. The real or simulated results of the atmospheric TBP are an obligatory path between the experimental results and the generation of bases for calculation of thermodynamic and thermophysical properties for different cuts. 

Cutting petroleum into fractions is done by the method illustrated in Figure 4.2. The petroleum fractions should correspond to the characteristics described in article 4.1.2. If certain characteristics are estimated, it is mandatory to compare the calculational results with the properties of the cuts and to readjust the estimated characteristics. 

For non-polar components like hydrocarbons, the results are very satisfactory for calculations of vapor pressure, density, enthalpy, and specific, heat and reasonably close for viscosity and conductivity provided that is greater than 0.10. 

The constants k- enable the improved representation of binary equilibria and should be carefully determined starting from experimental results. The API Technical Data Book has published the values of constants k j for a number of binary systems. The use of these binary interaction coefficients is necessary for obtaining accurate calculation results for mixtures containing light components such as ... 

The results of the calculation are close to those of the laboratory for medium and heavy cuts. They are somewhat different for light cuts, notably those containing dissolved gases. 

The accuracy depends on the fraction distilled it deviates particularly when determining the initial and final boiling points the average error can exceed 10°C. When calculating the ASTM D 86 curve for gasoline, it is better to use the Edmister (1948) relations. The Riazi and Edmister methods lead to very close results when they are applied to ASTM D 86 calculations for products such as gas oils and kerosene. 

Sufficiently accurate thermodynamic models used for calculating these equilibria are not available In simulation programs. It Is generally not recommended to use the models proposed. Only a specific study based on accurate experimental results and using a model adapted to the case will succeed. 

Calculating the hydrate formation temperature is essential when one needs to guard against equipment and line plugging that can result when wet gas is cooled, intentionally or not, below 30°C. 

Table 4.23 Example calculation of the hydrate formation temperature for a natural gas at 80 bar abs. Result = 29.1 "C. ...

The VI is a number that results from a calculation involving the viscosities at 40°C and 100°C. It characterizes the capacity of the lubricant to maintain a constant viscosity through a large range in temperature. This property can be improved by additives. 

Calculational methods. Associating the analysis, the knowledge of the property-structure relationships, and the calculation methods has made possible the replacement of costly and arduous test methods by quicker tests whose results are linked by calculations to the characteristic under study. Some examples are the cetane number, in some cases, the octane number, or the characteristics of LPG (refer to Chapter 3). 

Showing the results as curves enables manual calculations to be made which are often useful in rough estimates. 

Below is a typical oil PVT table which is the result of PVT analysis, and which would be used by the reservoir engineer in calculation of reservoir fluid properties with pressure. The initial reservoir pressure is 6000 psia, and the bubble point pressure of the oil Is 980 psia. 

Connecting the measured points will result in a curve describing the area - depth relationship of the top of fhe reservoir. If we know the gross thickness (H) from logs we can establish a second curve representing the area - depth plot for the base of the reservoir. The area between the two lines will equal the volume of rock between the two markers. The area above the OWC is the oil bearing GRV. The other parameters to calculate STOIIP can be taken as averages from our petrophysical evaluation (see Section 5.4.). Note that this method assumes that the reservoir thickness is constant across the whole field. If this is not a reasonable approximation, then the method is not applicable, and an alternative such as the area - thickness method must be used (see below). 

A Monte Carlo simulation is fast to perform on a computer, and the presentation of the results is attractive. However, one cannot guarantee that the outcome of a Monte Carlo simulation run twice with the same input variables will yield exactly the same output, making the result less auditable. The more simulation runs performed, the less of a problem this becomes. The simulation as described does not indicate which of the input variables the result is most sensitive to, but one of the routines in Crystal Ball and Risk does allow a sensitivity analysis to be performed as the simulation is run.This is done by calculating the correlation coefficient of each input variable with the outcome (for example between area and UR). The higher the coefficient, the stronger the dependence between the input variable and the outcome. 

From the probability distributions for each of the variables on the right hand side, the values of K, p, o can be calculated. Assuming that the variables are independent, they can now be combined using the above rules to calculate K, p, o for ultimate recovery. Assuming the distribution for UR is Log-Normal, the value of UR for any confidence level can be calculated. This whole process can be performed on paper, or quickly written on a spreadsheet. The results are often within 10% of those generated by Monte Carlo simulation. 

One significant feature of the Parametric Method is that it indicates, through the (1 + K 2) value, the relative contribution of each variable to the uncertainty in the result. Subscript i refers to any individual variable. (1 + K ) will be greater than 1.0 the higher the value, the more the variable contributes to the uncertainty in the result. In the following example, we can rank the variables in terms of their impact on the uncertainty in UR. We could also calculate the relative contribution to uncertainty. 

Take an example of estimating gross rock volume, based on seismic data and the results of two wells in a structure (Fig. 7.2). The following cross-section has been generated, and a base case GRV has been calculated. 

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