Example back calculation

The fundamental problem involved is always one of data quality, and consequently the back analysis approach must be applied with care and the results interpreted with caution. Back analysis is of use only if the soil conditions at failure are unaffected by the failure. For example, back-calculated parameters for a first-time slide in stiff overconsolidated clays could not be used to predict subsequent stability of the sliding mass, since the shear strength parameters will have been reduced to their residual values by the failure. In such cases, an assumption of c = 0 and the use of a residual friction angle ip is warranted (Bromhead, 1992). If the three-dimensional geometrical effects are important for the failed slope under consideration and a two-dimensional back analysis is performed, the back-calculated shear strength will be too high and thus unsafe. 

Example 8 Calculation of Rate-Based Distillation The separation of 655 lb mol/h of a bubble-point mixture of 16 mol % toluene, 9.5 mol % methanol, 53.3 mol % styrene, and 21.2 mol % ethylbenzene is to be earned out in a 9.84-ft diameter sieve-tray column having 40 sieve trays with 2-inch high weirs and on 24-inch tray spacing. The column is equipped with a total condenser and a partial reboiler. The feed wiU enter the column on the 21st tray from the top, where the column pressure will be 93 kPa, The bottom-tray pressure is 101 kPa and the top-tray pressure is 86 kPa. The distillate rate wiU be set at 167 lb mol/h in an attempt to obtain a sharp separation between toluene-methanol, which will tend to accumulate in the distillate, and styrene and ethylbenzene. A reflux ratio of 4.8 wiU be used. Plug flow of vapor and complete mixing of liquid wiU be assumed on each tray. K values will be computed from the UNIFAC activity-coefficient method and the Chan-Fair correlation will be used to estimate mass-transfer coefficients. Predict, with a rate-based model, the separation that will be achieved and back-calciilate from the computed tray compositions, the component vapor-phase Miirphree-tray efficiencies. 

Knowing this ratio, it is a simple matter to back calculate to ascertain the length of time since the sample was last alive. For example, we know that a time of one half-life t /2 has elapsed if a sample contains exactly half the expected amount of 14C, so the sample died 5570 years ago. 

In this approach, the desired closedloop response for a given input is specified. Then, knowing the model of the process, the required form and tuning of the feedback controller is back-calculated. These steps can be clarified by a simple example. 

Methylmercury is a primary example of this type of model in humans. A human mercury one-compartment PK model has been developed on the basis of several exposure datasets in which blood or hair concentrations have been measured at various times after methyl mercury exposure (Stern 1997 Kershaw et al. 1980 Sherlock et al. 1984). The one-compartment model takes into account the elimination half-life of mercury from blood and methylmercury volume distribution (in blood) to back-calculate the... 

However, it is also common to use standards to set up the infrastructure, policies, controls, or rules that mean that incidents and risks occur with acceptably rare probabilities. These standards might be described as strategic standards. For example, controls on ammonia in sewage treatment works (which are back-calculated from environmental standards) are designed to promote good fisheries in the receiving river. The intention is to reduce serious incidents to an acceptable frequency in each river because the infrastructure of sewage treatment appears to function at this level of acceptable risk. This may result in a compromise, which is essentially that standards are set up as particular types of summary statistics and not as absolute limits. 

Example 11 Calculations of Gas Emissivity and Absorptivity Consider a slab of gas confined between two infinite parallel plates with a distance of separation of L = 1 m. The gas pressure is 101.325 kPa (1 atm), and the gas temperature is 1500 K (2240°F). The gas is an equimolar mixture of C02 and H20, each with a partial pressure of 12 kPa (pc = pw = 0.12 atm). The radiative flux to one of its bounding surfaces has been calculated by using RADCAL for two cases. For case (a) the flux to the bounding surface is 68.3 kW/m2 when the emitting gas is backed by a black surface at an ambient temperature of 300 K (80°F). This (cold) back surface contributes less than 1 percent to the flux. In case (b), the flux is calculated as 106.2 kW/m2 when the gas is backed by a black surface at a temperature of 1000 K (1340°F). In this example, gas emissivity and... 

As previously indicated, the determination of a drug concentrations in plasma specimens requires the construction of a calibration response curve. This curve is often constructed as a straight line from the measured peak response ratios (y,) plotted against their respective calibrator concentrations (x,). The drug concentration in a specimen or the apparent (back-calculated) calibrator concentration is obtained from a rearrangement of theequation for the calibration line (without error) x,- = (y,- — b)/m. An example of a calibration curve with back-calculated concentrations is given the first four columns of Table 10. 

To determine the actual concentration per nucleotide base, prepare a 1 20 dilution remove 50 xl from your DNA stock, dilute to 1000 xl and obtain a UV spectmm. The molar absorptivity per nucleotide base is e260 = 6600 m-1 cm-1. In this example, the A260 should be 0.845. Using Beer s law, the concentration of nucleotide base is 0.845 1 cm-1 -6600 m-1 cm-1 = 1.28 x 10-4 mol nucleotide l-1. In terms of mol base-pair l-1, (1.28 x 10-4/2) = 0.64 x 10-4 mol base-pair l-1. Back-calculating, the... 

Maintaining the converter in CCM, down to the minimum load of our application, enforces a certain maximum value for r. For example, if the minimum load is Io min = 0.5 A, then to maintain the converter in CCM at 0.5 A, the set current ripple ratio (r at 3 A) needs to be lowered. Back calculating, we get the required condition for this... 

The latter approach—completely drying the aerosol— while valid, is diffi-cnlt to perform experimentally (29). The low solids content of pharmaceutical nebulizer solutions results in dry aerosols that are too fine for reliable enough measurements to be made to allow back calculation with any accuracy. For example a typical nebulizer droplet distribution with a mass median diameter of 3 pm generated from a 0.1% drug solution would result in a dry particle distribution with a mass median diameter of 0.1 pm. Obtaining size distribution data with any reasonable resolution at this size is very problematic. A further issue is that the solids content of the original droplets must be known in order to perform the calculation, and this can vary considerably during the course of nebulization. This technique has therefore not been used extensively. 

Equations fl6-86bl and tl6-921 are not identical because they are based on different models. Both models assume that the continuous phase is well mixed, but their assumptions about the dispersed phase are different. These differences emphasize the comments in Chapter 15 that analysis of mass transfer remains a topic for research and discussion. With identical values of KQ.E a, Eq. fl6-921 always predicts a higher value for E than does Eq. n6-86bl. This is illustrated in Example 16.5. Another way of looking at this is that with the same E, a larger KQ.E a value will be back-calculated from Eq. (16-... 

When a decrease of the original volume (via solvent extraction, distillation, or evaporation, etc.) is required before analysis, back calculations must be carried out to determine the original volume that must be collected to give the desired measurable level of the parameter. For example, if Vj is the volume calculated from the use of Ms and R, then the required original volume Vq is obtained from the product of Vi and the volume reduction factor For a volume reduction of 100, Vo would be lOOVi. 

Calibration curve data should be assessed to determine the appropriate mathematical regression that describes the instrument s response over the range of thecalibration curve (Section8.5). The report should include the back-calculated concentration values, accuracies, slopes, y-intercepts and correlation coefficients (R) and the coefficients of determination (R ) (Equation[8.18] in Section 8.3.1) for aU curves used in the validation. The value should be > 0.98 for each calibration curve. The R value (if used) must be > 0.99 for each calibration curve. An example table used to summarize the calibration curve statistics for each run used for method validation is shown as Table 10.2. 

The back side here the results of the checks and final evaluation are listed (evaluation phase). There is also a small blank space, for example for calculations or remarks. 

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