Cumulative curve

A plot of the last entry versus M gives the integrated form of the distribution function. The more familiar distribution function in terms of weight fraction versus M is given by the derivative of this cumulative curve. It can be obtained from the digitized data by some additional manipulations, as discussed in Ref. 6. 

Figure 2.2 shows the cash flow pattern for a typical project. The cash flow is a cumulative cash flow. Consider Curve 1 in Figure 2.2. From the start of the project at Point A, cash is spent without any immediate return. The early stages of the project consist of development, design and other preliminary work, which causes the cumulative curve to dip to Point B. This is followed by the main phase of capital investment in buildings, plant and equipment, and the curve drops more steeply to Point C. Working capital is spent to commission the plant between Points C and D. Production starts at D, where revenue from sales begins. Initially, the rate of production is likely to be below design conditions until full production is achieved at E. At F, the cumulative cash flow is again zero. This is the project breakeven point. Toward the end of the projects life at G, the net rate of cash flow may decrease owing to, for example, increasing maintenance costs, a fall in the market price for the product, and so on.

The cumulative curve obtained from the transit time distribution in Figure 9 was fitted by Eq. (48) to determine the number of compartments. An additional compartment was added until the reduction in residual (error) sum of squares (SSE) with an additional compartment becomes small. An F test was not used, because the compartmental model with a fixed number of compartments contains no parameters. SSE then became the only criterion to select the best compartmental model. The number of compartments generating the smallest SSE was seven. The seven-compartment model was thereafter referred to as the compartmental transit model. 

The mean diameters d of the smaller diaimeter distributions calculated from the cumulative curves are about 0.1 vn greater than those calculated algebraically. Because the higher dicutieter distributions contain no more than 3% of the particles counted (and sometimes less than 0.05%), no attempt was made to resolve the two distributions present. Applying a correction for the particles in the upper distribution would have a negligible effect on d and a calculated from the cumulative curve for the smaller diameter distribution. 

Equation (11.16) represents a convenient means of reducing the cumulative curve to the distribution curve which gives the pore volume per unit radius interval. 

Figure 11.8 is the distribution prepared from Fig. 11.7. A series of values of the ratio AF/AP is taken from Fig. 11.7 or preferably from the raw data. Each value of AF/AP is multiplied by the pressure at the upper end of the interval and divided by the corresponding pore radius. Alternatively, the mean pressure and radius in an interval may be used to calculate D (r). The resulting D (r) values are plotted versus the pore radius. Table 11.1 contains the raw data from which the cumulative curve. Fig. 11.7, is obtained, as well as the tabularized calculations necessary to prepare the distribution curve. Fig. 11.8.

Plots of the derivative of the cumulative curves, dV/dP versus pressure or radius, are often useful for the determination of the radius or pressure at which the maximum volume intrudes or extrudes. Figures 11.14a and 11.14b are derivative plots calculated from the cumulative curves in Figures 11.9a and 11.9b. 

Another way in which these kinds of data are sometimes represented is as a cumulative curve in which the total number (or fraction) of particles nT>, having diameters less (sometimes more) than and including a particular d, are plotted versus dr Figure 1.18b shows the cumulative plot for the same data shown in Figure 1.18a as a histogram. The cumulative curve is equivalent to the integral of the frequency distribution up to the specified class mark. Cumulative distribution curves are used in Chapter 2 in connection with sedimentation. 

Solution These cumulative percentages are of the same form as in Figure 2.5 therefore, the particle size distribution peaks where the cumulative curve increases most steeply. This occurs at about 0.29 for this distribution. Equation (10) permits the sedimentation velocity for particles of this size to be calculated ... 

As far as figure 13.5 is concerned, it should be noted that the percentage along the y axis only applies to the cumulative curve. 

Fig. 13.5 Histogram of the mass percentage of particles as a function of the particle diameter and the accompanying cumulative curve.

When variability and uncertainty are propagated separately (e.g. by two-dimensional or 2D Monte Carlo), they can be shown separately in the output. For example, the output can be presented as three cumulative curves a central one representing the median estimate of the distribution for variation in exposure, and two outer ones representing lower and upper confidence bounds for the distribution (Figure 2). This can be used to read off exposure estimates for different percentiles of the population, together with confidence bounds showing the combined effect of those uncertainties that have been quantified. 

Analysis of experimental human small-intestine transit time data collected from 400 studies revealed a mean small-intestinal transit time (TSi) = 199 min [173]. Since the transit rate constant kt is inversely proportional to (TSj), namely, kt = to/ (TSi), (6.12) was further fitted to the cumulative curve derived from the distribution frequency of the entire set of small-intestinal transit time data in order to estimate the optimal number of mixing tanks. The fitting results were in favor of seven compartments in series and this specific model, (6.10) and (6.11) with to = 7, was termed the compartmental transit model. 

Plot a cumulative curve of area versus lignin concentration. 

Divide the cumulative curve into regions of secondary wall and middle lamella on the basis of lignin concentration. 

Determine the average lignin concentrations in the two regions from the cumulative curve. 

Unfortunately, most emulsions do not have a single droplet size. There are small, medium and large droplets present, and it is important to be able to characterise the emulsion for this. This is done by counting the number of particles that is smaller than a specific size, for many different sizes. The resulting data can then be plotted on a curve, the cumulative distribution curve. Alternatively, one can count all particles that have a size within an interval of sizes (e.g., 1-2 pm), and do this for all intervals. Plotting all the numbers obtained for all intervals, then results in a frequency distribution. The two distributions are closely related the derivative of the cumulative curve to the particle size, will give a (continuous) curve that is similar to the discrete frequency distribution obtained earlier, and the smaller the intervals are chosen, the closer the derivative will follow the frequency distribution (see Figure 15.4). 

A frequency distribution curve can be used to plot a cumulative-frequency curve. This is the curve of most importance in business decisions and can be plotted from a normal frequency distribution curve (see Sec. 3). The cumulative curve represents the probability of a random value z having a value of, say, Zi or less. 

Figure 20-1 presents the data from Table 20-1 in both cumulative and frequency format. In order to smooth out experimental errors it is best to generate the frequency curve from the slope of the cumulative curve, to use wide-size intervals or a data-smootning computer program. The advantage of this method of presenting frequency data is that the area under the frequency curve equals 100 percent, hence, it is easy to visually compare similar data. A typical title for such a presentation would be Relative and cumulative mess distributions of quartz powder by pipet sedimentation. 

Fig. 2a. Precrosslinked poly(organosiloxane) particles (5 mol% T units) before grafting of PMMA (degree of grafting 50 %) transmission electron micrographs, volume distributions, cumulative curves...

Figure 4.14. Particle Size Distributions comcspoixJing to a Rh(2.4%VCe02 catalysts submitted to reduction treatments at increasing temperatures in the 623 - 1173 K range and cumulative curves showing percentage of particles versus diameter obtained for each reduction temperature (I S3).

Figure 3 Cumulative curve of weight of acid-insoluble fraction ( clay ) as a function of time (BP) in an Atlantic equatorial core from the Mid-Atlantic Ridge (LDEO core A 180-74) (source Broecker et al, 1958).

A steady state natural population, characterized by a continuous generation/destruction (birth/ death) cycle, is usually typified by an age structure similar to that in Figure 1, the cumulative curve defining all necessary parameters of a given population. These are its half-life mean age hnean> d obUvion age or life expectancy... 

Figure 15.1. Cumulative curves showing the frequency distribution of various constituents in terrestrial water. Data are mostly from the United States from various sources. (Adapted from Davies and DeWiest, 1966.)... 

Figure 5 shows the differential distributions versus pore size for the two sets of data. They are obtained by numerical differentiation of cumulative curves given on figure 4. They confirms more finely the identity of pore size distributions obtained fi-om the two different curves by the adequate interpretation. 

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