Operating lines for four different absorbent flow rates are shown in Fig.
The value of the solute concentration in the outlet liquid, X N , depends on l 0 by a material balance on the solute for the entire absorber. Note that the operating line can terminate at the equilibrium line, as for operating line 4, but cannot cross it because that would be a violation of the second law of thermodynamics. It takes an infinite number of derived from 12 as follows. For stage N, 3 becomes:. The selection of the actual operating absorbent flow rate is based on some multiple ofL min , typically from 1.
A value of 1. In Fig. As the operating line moves form 1 to 4, the number of required equilibrium stages, N, increases from zero to infinity. Thus, a trade-off exists between L and N , and an optimal value of L exists Suleimandiab, and Walas, In many instances, absorption is carried out in a packed column, which may be considered to be a differential contactor. For the contactor shown in Fig. The interfacial area available for mass transfer is a function of the type and size of packing used in the tower. By incorporating a characteristic of the packing what is an indication of the interfacial contact area, we have:.
The mass transfer that occurs in the differential height in be expressed in terms of either the change in the gas-phase composition or the change in the liquid phase composition. An order to use equation 21 , one must be able to relate partial pressure and composition in the gas phase, or concentration and composition in the gas phase, or concentration and composition in the liquid phase. For the gas phase, we can relate the partial pressure and composition by:. In order to obtain the height of packing required in a given application, equation 24 must be integrated:.
The relationship between bulk and interfacial compositions is, in general, not easily expressible in simple fashion that will allow for analytical integration. For this reason, equation 25 must ordinarily be integrated graphically or numerically Shulman and Margolis, ; McNulty and Chopey, There are two ways in which the total column required for making a given solute recovery can be estimated. One is through the use of the height equivalent to a theoretical plate H. The H. The same thing is true of the H. Many feel that for a packed column, which approximates a differential contactor, use of the H.
The number of equilibrium stages, N, based on vapor-phase mass transfer and with considering of k n be constant is given by:. A number of methods have been suggested for estimating mass-transfer coefficients for absorption. All of these tend to be both system and packingtype dependent. As a result, few have wide-range general applicability, and all must be used with considerable caution. By far the best procedure is to obtain experimental data for the specific system and packing type being considered.
Liquid and gas film coefficients can be calculated by the equations as follows Chhabra, et al. The overall height of a transfer unit can be defined as the height of a packed section required to accomplish a change in concentration equal to the average driving force in that section. Values of HTU OG for a particular system are sometimes available directly from the literature or from pilotplant tests, but often they must be estimated from empirical correlations for the individual coefficients or the individual heights of a transfer unit.
Overall gas:. A histogram is constructed by dividing the range of data values into equal intervals on the abscissa axis and then placing a bar over each interval of a height equal to the number of data points within that interval. The number of intervals should be few enough so that almost all intervals contain at least three points, however there should be a sufficient number of intervals to facilitate visualization of the shape and symmetry of the bar heights.
Twenty intervals are usually recommended for a histogram. Because so many points are required to construct a histogram, it is unusual to? This distribution is bellshaped and symmetrical.
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Use of analysis techniques developed for the normal distribution on data distributed in another manner can lead to grossly erroneous conclusions. Thus, before attempting data analysis, the data should either be veri? Transformed data may be analyzed statistically and the results transformed back to give the desired results, although the process of transforming the data back can create problems in terms of not having symmetrical con?
This paper has one linear axis, and one axis which is arranged to re? The marks are spaced far apart in the center and close together at the ends. A normal probability plot may be constructed as follows with normal probability paper. NOTE 1—Data that plot approximately on a straight line on the probability plot may be considered to be normally distributed. Deviations from a normal distribution may be recognized by the presence of deviations from a straight line, usually most noticeable at the extreme ends of the data.
The numbers of the data points thus obtained are called the ranks of the points.
NOTE 2—Occasionally two or more identical values are obtained in a set of results. In this case, each point may be plotted, or a composite point may be located at the average of the plotting positions for all the identical values. This is the area to the left of that value under the standardized normal distribution. The cumulative distribution function is the number, always between 0 and 1, that is plotted on the probability axis.
Although it is sometimes possible to guess at the type of distribution by looking at the histogram, and thus determine the exact transformation to be used, it is usually just as easy to use a computer to calculate a number of different transformations and to check each for the normality of the transformed data. Some transformations based on known non-normal distribu- where: y 5 transformed datum, x 5 original datum, and n 5 number of data points. Time to failure in stress corrosion cracking usually is best? Once a set of transformed data is found that yields an approximately straight line on a probability plot, the statistical procedures of interest can be carried out on the transformed data.
Results, such as predicted data values or con? If this distribution is not normal, a transformation may be sought which will normalize that particular distribution.
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See 4. Analysis may then be conducted on the transformed data.
Non-parametric tests do not use the data as efficiently. In this case, Guide G 46 should be consulted for the procedure 4. See Refs.
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Sometimes mistakes can be identi? Central Measures 5. If it is assumed that the processes operating to create error in the measurement are random in nature and are 2 4 The boldface numbers in parentheses refer to the list of references at the end of this guide. G 16 as likely to overestimate the true unknown value as to underestimate it, then the average value is the best estimate of the unknown value in question. The average value is usually indicated by placing a bar over the symbol representing the measured variable.
A quali? In cases of penetration from pitting and cracking, failure is often de? Extreme value analysis has been used in these cases, see Guide G Variability Measures 6. In the case of normal distribution, a number of procedures are available and can be handled with computer programs. These measures include the following: variance, standard deviation, and coef?
The range is a useful non-parametric estimate of variability and can be used with both normal and other distributions. When reporting standard deviation calculations, it is important to note clearly whether the value reported is the standard deviation of the average or of a single value. In either case, the number of measurements should also be reported.
The sample estimate of the standard deviation is s. This measure of? The range is non-parametric in nature, that is, its calculation makes no assumption about the distribution of error. In cases when small numbers of replicate values are involved and the data are normally distributed, the range, w, can be used to estimate the standard deviation by the relationship: S.
Variance is a useful measure because it is additive in systems that can be described by a normal distribution, however, the dimensions of variance are square of units. A procedure known as analysis of variance ANOVA has been developed for data sets involving several factors at different levels in order to estimate the effects of these factors.
See Section 9. It has the property of having the same dimensions as the average value and the original measurements from which it was calculated and is generally used to describe the scatter of the observations. The range has the same dimensions as standard deviation. A tabulation of the relationship between s and w is given in Ref. The standard deviation of the error of measurement may be used as a measure of imprecision. This aspect is sometimes called repeatability. This aspect is sometimes called reproducibility. When applied to individual observations, bias includes a combination of a random component and a component due to systematic error.
Under these circumstances, accuracy contains elements of both precision and bias. Bias refers to the tendency of a measurement technique to consistently under- or overestimate. In cases where a speci? This is particularly true for test procedures which produce damage rapidly as compared to the service experience. In such cases, it is important to establish the correspondence between results from the service environment and test results for the class of material in question.
Bias in this case refers to the variation in the acceleration of corrosion for different materials. Bias in this type of test refers to the inability of the test to properly rank the materials to which the test applies as compared to service results. Ranking may also be used as a qualitative estimate of bias in the test method types described in 6.
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