Trendline

Figure 4.16 (left). Trendlines for the various compwnents. The three scales are different %, ppm, resp. %). (right). Total impurities (columns 1-6, including water of crystallization, versus the HPLC assay of the major compound (column 7). The circle marks the hypothetically pure compound 3.2% water of crystallization, but no other impurities. The arrow indicates the percentage of impurities expected (for this simple linear model) to remain in the product after all solvents and excess water have been driven off. 

In Fig. 4.39, results for spiked placebo and for the verum tablets are given for compound A (bold lines) and B all horizontal bars should be at 100%, and the vertical lines should be centered at the same height. The gray trendlines, particularly for the LO- and Hl-range A-values indicate a systematic difference in response between tbe calibration solutions and the spiked placebo tablets (extraction efficiency, interference, etc.). For same ranges, the verum-tablets assays either underestimate the content of A by 4—5%, or A is underdosed. For compound A the repeatability figures are as follows (%-of-nom-inal, see file Fig4 39.dat), see Table 4.36. 

In this study the reader is introduced to the procedures to be followed in entering parameters into the CA program. For this study we will keep Pm = 1.0. We will first carry out 10 runs of 60 iterations each. The exercise described above will be translated into an actual example using the directions in Chapter 10. After the 10-run simulation is completed, determine (x)6o, y)60, and d )6o, along with their respective standard deviations. Do the results of this small sample bear out the expectations presented above Next, plot d ) versus y/n for = 0, 10,20, 30,40, 50, and 60 iterations. What kind of a plot do you get Determine the trendline equation (showing the slope and y-intercept) and the coefficient of determination (the fraction of the variance accounted for by the model) for this study. Repeat this process using 100 runs. Note that the slope of the trendline should correspond approximately to the step size, 5=1, and the y-intercept should be approximately zero. 

The parameters are easily determined by using computer software. In Microsoft Excel, the data are put into columns A and B and the graph is created as for a linear curve fit. This time, though, when adding the trendline, choose the polynomial icon and use 2 (which gives powers up to and including x ). The result is... 

Whereas the effect of water on deactivation and on the overall activity of the FTS varies with the support, similar effects of water on the selectivity is reported for all catalysts, to a certain degree independent of the support, promoter and conditions. The effect can be summarized as an increase in C5 + selectivity, a decrease in methane selectivity, and in some instances a weak enhancement of the C02 selectivity is observed. Fig. 4 illustrates the effect on the C5 + and methane selectivity of adding water to cobalt catalysts supported on alumina, silica and titania, and both unpromoted and Re-promoted catalysts are shown. At the outset these selectivities are strong functions of the conversion, the C5 + selectivity increasing and the methane decreasing with increasing conversion, as illustrated by the trendlines in the figures. The points for methane are below, and C5 + -selectivity is above the line when water is added. Similar results were reported by many authors for alumina-supported catalysts,16-19 23 30 silica-supported catalysts,30 37 46-48 and titania-supported catalysts.19 30... 

Type in a chart title and labels for the x- and y-axes. Click finish. With either the graph or the entire graph box highlighted, click chart (upper tool bar), then add trendline, then OK. ... 

Rather than writing a short program in Matlab for this result, we demonstrate how to perform the task of a straight line fit in Excel. Excel actually provides several ways of performing the job of fitting the best line through a set of data pairs. The most convenient is probably the Add Trendline. .. tool which delivers the result in a few clicks. 

Select Add Trendline. .. to get the graphical input selection menu for the trendline as shown in Figure 4-8. 

One difficulty with the Trendline is that the equation only appears graphically. The values for slope and intercept have to be copied manually into the spreadsheet if they are to be used in later calculations. 

Excel has an alternative way of doing the same. The column of l s can be omitted, selecting only the three others and fitting with the LINEST function but having the Const option set to TRUE. Excel internally adds the columns of ones and delivers exactly the same results. That one special parameter is the z/-intercept. Similar options exist in the trendline function. 

Bill Excel LINEST function. Enter the data from Problem 4-23 in a spreadsheet and use the LINEST function to find the slope and intercept and standard deviations. Use Excel to draw a graph of the data and add a TRENDLINE. 

PROBLEM The following table presents V-[S] data. The units for V are fiM/s, and the units for [S] are mM. Plot the data points first as V versus [S] and then as I / V versus 1 / [ S ]. Include a trendline with each graph. Where do the data points cluster in each graph ... 

SOLUTION The plot for these points is shown in the following figure with a trendline and its best-fit equation. The graph is a form of Figure 3.18, and the equation we are trying to match is Equation 4.17. 

Figure 3 shows the crude runs necessary to meet our refined product demand forecast. The forecast includes two scenarios one of normal trendline growth portrayed by several industry observers and Pacefs own outlook for refinery crude runs based on a "cyclical11 economic forecasting model. Our forecast shows runs to crude stills will remain below 12.5 million barrels per day for the remainder of the decade. 

Now that everyone has a powerful computer in their back-pack or on their desk, it is simple to fit a straight or even a curved line to data using the TrendLine feature of Excel. As an example, let us work on some real data for DDT in trout from Lake Michigan. 

Figure 2.3 Graph created with Microsoft Excel showing the natural logarithms of the concentrations of DDT in trout from Lake Michigan (see the above table) as a function of time and showing a fitted straight line (using the TrendLine feature). The negative slope of this line is the rate constant.

Strategy. Let us just plot the raw data in Excel and use the TrendLine feature to fit an exponential line. Be sure to turn on the optional show equation and correlation coefficients feature. We get the plot shown in Figure 2.5. 

Fig. 8.4 shows industrial catalytic converter (hence catalyst bed) diameters as a function of measured 1st catalyst bed feed gas volumetric flowrates. Bed diameters are between 8 and 16 m. They increase with increasing input gas flowrate. They are quite precisely predicted by the trendline equation on the graph. 

Fig. 2. Relative activation energy for the reaction between MnO and UOj as function of the UOj particle size. The trendline was calculated using Eq. (4).

Figure 1.10 (A) Plot of NO versus PO) from all depths and from selected WOCE cruises in all ocean basins (A16,P16,I8NI9S). The inset shows how the various processes influence the NO versus PO) distribution. The solid line represents the mean ocean trend with a slope of 16 1, while the thinner lines show trends of constant N (see text for deflnition). (B) Hypothetical distribution of NOy versus PO) in a situation of excess NOy. (C) as (B), except for a situation of a NOy deficit. It is unclear why the intercept of the mean oceanic trendline in (A) is so close to zero. From Gruber (2004).

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