42 Effect analyses of I-I, I-O and PI-O parameters
42.3 DP/IP analysis of I-I, I-O and PI-O parameters
DP/IP (direct proportionality/inverse proportionality) analysis was introduced to be able to understand direct/inverse proportionality of input and output parameters of interaction of laser beam and paper material.
As linear trend line was fitted to parameters data points, an equation of linear dependence for each parameter combinations was defined. This equation was determined by Excel spreadsheet program and an option of linear line fitting and corresponding equation was chosen. General equation of linear line is shown in equation 42.1.
y = kx + b (42.1)
Equation 42.1 can be written also as function as equation 42.2 presents.
f(x) = kx + b (42.2)
As x is set to be zero x = 0, point where fitted linear line crosses y-axis, namely f(0), can be defined as equation 42.3 illustrates.
f(0) = b (42.3)
Direct/inverse proportionality can be defined, when these f(0) values are calculated to each parameter combinations shown in tables 42.3 and 42.5 and they are compared to each other.
Appendix 12 shows an example of this calculation and DP/IP analysis.
Table 42.6 shows the system of marking direct/inverse proportionality.
Table 42.6 Marking of direct/inverse proportionality.
Proportionality Sign Direct proportionality Inverse proportionality No proportionality * 42.4 Correlation analysis of I-I, I-O and PI-O parameters
Correlation analysis was carried out to be able to define which input and output parameters coming from DP/IP analysis have the best correlation between each other i.e. which parameters have significant effect on each other.
Correlations were defined by Excel spreadsheet program by fitting linear trend line to parameter data points shown in table 42.1. When fitting trend lines an option of linear line fitting and R2 value representing correlation was chosen.
Correlation determination was carried out to each parameter combinations shown in tables 42.3 and 42.5. Correlations were compared and evaluated with a system shown in table 42.7.
Table 42.7 Correlations and their descriptions.
Correlation R2, - Description R2 < 0.10 Very poor, insignificant 0.10 < R2 < 0.30 Poor 0.30 < R2 < 0.45 Rather poor 0.45 < R2 < 0.55 Adequate 0.55 < R2 < 0.70 Satisfactory 0.70 < R2 < 0.90 Good
R2 > 0.90 Significant
From all correlations the ones reaching the level of adequate, satisfactory, good and significant (R2
> 0.45) were chosen for further analysis. All correlation values higher than 0.70 represented in tables 42.3 and 42.4 are also taken into account in next analysis of dependence because of their high correlation value.
42.5 Dependence analysis of I-I, I-O and PI-O parameters
Dependence analysis was executed for parameters resulting from correlation analysis to be able to understand the dependence of input and output parameters of interaction of laser beam and paper material and its characteristics.
As linear trend line was fitted to parameters data points shown in tables 42.3 and 42.4, also corresponding equation was determined. It can also be written as function as equation 42.1 illustrates. When this function is derivated, f´(x) can be defined as equation 42.4 represents.
f´(x) = k (42.4)
Derivate tells how function changes as x changes and it can be used to describe whether a function is increasing or decreasing. Table 42.8 shows the how increasing or decreasing characteristics of function is defined.
Table 42.8 Definition of increasing and decreasing characteristics of function Definiton Description
f´(x) > 0 Strictly increasing f´(x) < 0 Strictly decreasing f´(x) = 0 Constant
When these f´(x) values are calculated for each parameter combinations shown in tables 42.3 and 42.5, dependence can be defined. Appendix 13 shows example of this calculation and dependence analysis.
As table 42.8 shows increasing and decreasing characteristics of f´(x) is defined by value of zero.
However, it was noticed during experiments, that definition of exact value of zero is very difficult especially with experimental data. This is why an approximate value of 0.001 was used instead of exact value of zero. Table 42.9 shows the system used for marking of increasing/decreasing characteristics of functions used in this thesis.
Table 42.9 Definition of increasing/decreasing behaviour of functions to be able to define dependence between input and output parameters.
Definition Description Sign f´(x) > 0.001 Increasing f´(x) < -0.001 Decreasing -0.001 < f´(x) < 0.001 Constant
Ranking in dependence analysis is executed by evaluating if one parameter correlation has same increasing/decreasing behaviour inside series. If this behaviour is changing, it is concluded that there is no dependence and this parameter combination is not analysed further. But if parameter combination inside series has same increasing/decreasing behaviour, it will be included for discussion analysis. Table 42.10 shows principle of this ranking.
42.6 Diagram of effect analyses of I-I, I-O and PI-O parameters
Finally, ranking factors of effect analysis can be added to figure 42.1 and completed diagram of these analyses is illustrated in figure 42.2.
Figure 42.2. Diagram of effect analyses with ranking factors.
43 Effect analysis of BHR100 and BHR86
To be able to analyse interaction between laser beam and paper material (see figure 34.1) and to understand effect of input-output parameters to the whole process, BHR100 and BHR86 were analysed.
Effect analyses of BHR100 and BHR86 consisted only of correlation analysis. DP/IP analysis and dependence analysis cannot be executed since the dependence between I-I parameters and BHR100 and BHR86 obeys natural logarithm function and zero point of natural logarithmic function is undefined. Also zero point of first derivate is undefined. Fluence was determined to be used as I-I parameter, since it combines all three I-I parameters which are laser power, pulse length and focal plane position to one value. Detailed analysis system to study I-I parameters and BHR100/BHR86 is described in table 43.1.
Table 42.10 Ranking in dependence analysis.
I-O parameters Maximum
spectral
intensity Conclusion Ranking
Maximum
temperature Conclusion Ranking
f´(x) D f´(x) D
I-I parameters Fluence as
function
No effect of I-I parameter to I-O parameter (marked with ): 0.0001 < f´(x) < -0.0001 Increasing effect of I-I parameter to I-O parameter (marked with ): f´(x) > 0.0001 Decreasing effect of I-I parameter to I-O parameter (marked with ): f´(x) < -0.0001
Parameter group
Table 43.1 Analysis system of I-I parameters vs. BHR100/BHR86
Finally, ranking factors of effect analyses of BHR100 and BHR86 can be added to figure 42.1 and completed diagram of these analyses is illustrated in figure 43.1.
Figure 43.1. Diagram of effect analyses with ranking factors.
44 MMM analyses for quality parameters
It was decided to carry out MMM (minimum-median-maximum) analyses for parameters describing the quality of result in material after interaction of laser beam and paper material. These so called quality parameters are:
- hole area
- HAZ
- ΔHAZ - conicality.
Aim of MMM analyses is to find minimum range, median range and maximum range of quality parameter representing both ends of worst quality (minimum range and maximum range) as well as average quality (median range).
MMM analysis is executed by creating frequency distribution (or so called histogram) curve of hole area, HAZ, ΔHAZ and conicality results. Frequency distribution is used in mathematical statistics
BHR100
for describing empirical data (Råde and Westergren, 1995; Hayter, 2002). When this distribution curve is defined also minimum, median and maximum range of each parameter can be determined.
44.1 Determination of frequency distribution curve and its shape
Before it is possible to define minimum, median and maximum range of quality parameters, shape of the distribution curve has to be explored. According to statistical mathematics, frequency distribution is used to describe distribution of empirical data. If shape of frequency distribution curve is close to normal distribution, minimum, median and maximum ranges can be defined (Råde and Westergren, 1995; Hayter, 2002).
Frequency f is defined as number ni of event i in occurred in total number N of events for example in experimental data, as equation 44.1 shows (Råde and Westergren, 1995; Hayter, 2002).
ni
f (44.1)
where f frequency, -
ni number of event i, -.
In statistics of experimental data, class intervals of certain data set are very often defined, so actually frequency f of class interval is the number of numbers in the list that belong to the class interval i.e. frequency f is number ni of event i in specific class interval. These class intervals are chosen such that they evenly cover whole data set (Råde and Westergren, 1995; Hayter, 2002).
In statistics, relative frequency fi is often used to describe empirical probability and it is defined as absolute frequency normalized by total number of events and is calculated as equation 44.2 shows (Råde and Westergren, 1995; Hayter, 2002).
Relative frequency fi can be plotted for all events i as frequency distribution in histograms.
Appendix 14 introduces example of this definition of shape of distribution curve (Råde and Westergren, 1995; Hayter, 2002).
44.2 Definition of minimum, median and maximum ranges of quality parameters It was concluded to define statistically values of minimum, median and maximum ranges of data to be able to analyse experimental results.
Sample statistics provides numerical summary measures of a data set. To be able to define minimum, median and maximum areas of experimental data sets in this thesis sample percentiles were used. The pth percentile of sample is a value that has proportion p of the sample taking values smaller than it and a proportion 1-p taking values larger than it. So it is an estimate of pth percentile of distribution of sample observations (Råde and Westergren, 1995; Hayter, 2002).
In this thesis, it was decided to use 10 percentile of sample as definition of minimum range.
Respectively, 90th percentile of sample was used as definition of maximum range. Range of 45th -55th percentile of sample was concluded to be defined as median values. Appendix 15 shows an example calculation for defining minimum range, median range and maximum range (Råde and Westergren, 1995; Hayter, 2002).
44.3 I-I parameter analysis of minimum, median and maximum ranges of quality parameters
When minimum, median and maximum ranges of each quality parameter are defined, I-I parameter analysis to these ranges are executed, as table 44.1 shows.
Table 44.1 I-I parameter analysis for minimum, median and maximum range of quality parameters.
Minimum range Median range Maximum range I-I
This testing eliminates those minimum, median and maximum range values of each quality parameter values that are single, odd and out of majority of values. Discussion analysis is carried out for those quality parameters ranked with table 44.1.
44.4 Diagram of MMM analysis
Finally, ranking factors of MMM analyses can be added to figure 42.1 and completed diagram of these analyses is illustrated in figure 44.1.
45 Discussion analysis
Aim of discussion analysis is to further analyse results of effect analyses of I-I, I-O and PI-O parameters, effect analyses for BHR100/BHR86 and MMM analyses, such that conclusions can be drawn and phenomena of interaction of laser beam and paper material can be understood.
45.1 Discussion analysis for results of effect analysis of I-I, I-O and PI-O parameters This part consists of detailed analysis of figures belonging to different parameter combinations coming out from effect analysis. This enables conclusions to be drawn and characteristics of laser beam and paper material understood.
45.2 Discussion analysis for results of effect analysis of BHR100 and BHR86
This part consists of determination of BHR100 /BHR86 limit fluence. These are limit values where curves fluence vs. BHR100 and fluence vs. BHR86 turns from steeply decreasing characteristics of curve into slowly decreasing characteristics of curve, and value of BHR100 and BHR86 gradually approaches zero. It was noticed that fluence vs. BHR100 and fluence vs. BHR86 follows general form of equation which is shown in equation 45.1.
y = - k ln x + b (45.1)
Figure 44.1. Diagram of MMM analyses with ranking factors.
Equation 45.1 can be written also as function as equation 45.2 presents.
f(x) = - k ln x + b (45.2)
First derivate of equation 45.2 is now calculated, as equation 45.3 illustrates.
f´(x) = - k (45.3)
x
It was noticed from derivated functions of fluence vs. BHR100 and fluence vs. BHR86 curves that as x approaches value of -0.1, also limit fluence value starts to approach zero, and this is why limit BHR100 limit fluence and BHR86 limit fluence is calculated as equation 45.4 represents. Appendix 16 shows example of this calculation.
f´(x) = - k (45.4)
x -0.1 x
45.3 Discussion analysis of MMM analyses
When it is analysed which input parameters were used to achieve these ranges and actually how was quality looking in micrographs, it can be concluded if certain parameters really is good one to describe quality of laser treatment, how good quality was achieved and which monitor device is suitable to analyse this good quality. Appendix 17 shows definition for this system of visual evaluation. Visual quality of laser treated dried pulp was evaluated by:
- overall quality,
III RESULTS
46 Effect analyses of I-I, I-O and PI-O parameters
To be able to understand characteristics of interaction between laser beam and paper material (see figure 34.1) and to conclude effect of I-I, I-O and PI-O parameters to whole process, effect analyses were executed. In this thesis, effect analysis means:
- DP/IP (direct proportionality/inverse proportionality) analysis,
- this analysis is carried out to be able to characterise direct/inverse proportionality of input and output parameters.
- correlation analysis and
- this analysis is carried out to be able to determine which input and output parameters have best correlation between each other.
- dependence analysis.
- this analysis is done to be able to study characteristics of input and output parameters.
46.1 DP/IP analysis
Table 42.2 and 42.3 introduces concept of DP/IP (direct proportionality/inverse proportionality) analysis of effect of I-I parameters to I-O and PI-O parameters. This analysis finds out direct/inverse proportionality of different input and output parameters and helps to understand direct/inverse proportionality of each parameter to interaction.
46.1.1 DP/IP analysis of I-I and I-O parameters
First part of DP/IP analysis (see table 42.2) is carried out for I-I parameter (fluence) and I-O parameters (maximum spectral intensity and maximum temperature). All numerical data used for this analysis is also illustrated in appendix 18. This analysis is shown in appendix 19.
Table 46.1 introduces DP/IP analysis of I-I and I-O parameters.
As from table 46.1 can be seen, fluence as function of pulse length has direct proportionality.
46.1.2 DP/IP analysis of I-I and PI-O parameters
Table 42.2 shows the second DP/IP analysis consisting of analysis of I-I parameter (fluence) and PI-O parameters (hole area, HAZ, ΔHAZ and conicality). All numerical data used for this analysis is shown in appendix 20. This analysis is introduced in appendix 21.
Table 46.2 illustrates comparison of DP/IP of I-I and PI-O parameters.
Table 46.2 shows direct proportionality when following combinations are taken into account:
- fluence as function of average laser power vs. hole area, - fluence as function of pulse length vs. hole area,
- fluence as function of average focal plane position vs. hole area, - fluence as function of average laser power vs. ∆HAZ,
- fluence as function of average focal plane position vs. ∆HAZ, - fluence as function of average laser power vs. conicality and - fluence as function of pulse length vs. conicality.
Only fluence as function of pulse length vs. ∆HAZ has inverse proportionality.
Table 46.1 DP/IP analysis of I-I and I-O parameters.
I-O parameters
Only one or two data point available, no reliable trend line could be drawn: N/A No conclusion could be drawn: -
Table 46.2 DP/IP analysis of I-I and PI-O parameters.
PI-O parameters
Only one or two data point available, no reliable trend line could be drawn: N/A No conclusion could be drawn: -
46.1.3 DP/IP analysis of I-O and PI-O parameters
Table 42.5 represents diagram of DP/IP analysis of I-O and PI-O parameters. This analysis is also grouped into analysis of subgroups of average laser power, pulse length and average focal plane position. Detailed analysis is introduced in appendix 22. All numerical data used for this analysis is also shown in appendices 18 and 20.
Table 46.3 shows DP/IP analysis of I-O and PI-O parameters.
As from table 46.3 can be noticed following parameter combination has direct proportionality:
‐ maximum spectral intensity as function of average laser power vs. maximum temperature and
‐ maximum temperature as function of average focal plane position vs. HAZ.
Table 46.3 reveals that following parameter combination has inverse proportionality:
‐ maximum spectral intensity as function of pulse length vs. maximum temperature,
‐ maximum spectral intensity as function of average laser power vs. hole area,
‐ maximum spectral intensity as function of pulse length vs. hole area,
‐ maximum spectral intensity as function of average focal plane position vs. hole area,
‐ maximum temperature as function of average focal plane position vs. hole area,
‐ maximum spectral intensity as function of pulse length vs. HAZ,
‐ maximum temperature as function of average laser power vs. HAZ and
‐ maximum temperature as function of average pulse length vs. HAZ.
46.1.4 DP/IP analysis of PI-O parameters
Table 42.5 shows DP/IP analysis of PI-O parameters. This analysis is also grouped into analysis of subgroups of average laser power, pulse length and average focal plane position. Detailed analysis is introduced in appendix 23. All numerical data used for this analysis is also shown in appendix 20.
Table 46.4 introduces DP/IP analysis of PI-O parameters.
Table 46.1 shows that hole area as function of average laser power vs. HAZ has direct proportionality and hole area as function of pulse length vs. HAZ has inverse proportionality.
46.1.5 Conclusions of DP/IP analysis of I-I, I-O and PI-O parameters
Appendix 24 introduces conclusions of all DP/IP (direct proportionality/inverse proportionality) analysis of I-I, I-O and PI-O parameters illustrated in tables 46.1, 46.2, 46.3 and 46.4.
Based on appendix 24, I-I, I-O and PI-O parameter combinations having direct or inverse proportionality are introduced in table 46.5. These parameters go for next analysis of correlation.
46.2 Correlation analysis
Table 42.2 and 42.3 describes content of correlation analysis of I-I, I-O and PI-O parameters. This evaluation reveals correlations of different parameters and helps to understand characteristics of basic parameters to interaction of laser beam and paper material.
Table 46.3 DP/IP analysis of I-O and PI-O parameters.
I-O parameters PI-O parameters Maximum
temperature Hole area HAZ
f(0) f(0) f(0)
0.4 mm 409.690 -0.8480 -140.450
-0.6
Only one or two data point available, no reliable trend line could be drawn: N/A No conclusion could be drawn: -
46.2.1 Correlation analysis of I-I and I-O parameters
As can be seen from table 42.2, first part of correlation analysis is executed for I-I parameter (fluence) and I-O parameters (maximum spectral intensity and maximum temperature). All numerical data used for this analysis is also illustrated in appendix 18. This analysis is shown in appendix 19. Correlation analysis for all parameters is shown in appendix 25.
Table 46.6 illustrates comparison of correlation of I-I and PI-O parameters coming out from DP/IP analysis (see table 46.5).
Table 46.4 DP/IP analysis of PI-O parameters.
Only one or two data point available, no reliable trend line could be drawn: N/A No conclusion could be drawn: -
Table 46.5 I-I, I-O and PI-O parameter combinations resulting from DP/IP analysis and will be included for correlation analysis.
Fluence Maximum spectral intensity
Fluence Maximum temperature
Fluence Hole area
Fluence HAZ
Fluence ΔHAZ
Fluence Conicality Fluence as function of average laser power Hole area
Fluence as function of average laser power ΔHAZ Fluence as function of average laser power Conicality Fluence as function of pulse length Hole area
Fluence as function of pulse length ΔHAZ
Fluence as function of pulse length Conicality Fluence as function of average focal plane position Hole area Maximum spectral intensity as function of average laser power Hole area Maximum spectral intensity as function of average laser power HAZ
Maximum spectral intensity as function of pulse length Maximum temperature Maximum spectral intensity as function of pulse length Hole area
Maximum spectral intensity as function of pulse length HAZ
Maximum spectral intensity as function of average focal plane position Maximum temperature Maximum spectral intensity as function of average focal plane position Hole area
Maximum temperature as function of average laser power HAZ Maximum temperature as function of pulse length HAZ Maximum temperature as function of average focal plane position Hole area Maximum temperature as function of average focal plane position HAZ Hole area as function of average laser power HAZ Hole area as function of pulse length HAZ
Table 46.6 Correlation analysis of I-I and I-O parameters.
Correlation R2, - I-O parameters Maximum
spectral intensity Maximum temperature I-I
parameters Fluence - 0.2549 0.1661 Adequate range of R2(marked with italicfont): 0.45 < R2< 0.55
Satisfactory range of R2 (marked with green font): 0.55 < R2 < 0.70 Good range of R2 (marked with blue font): 0.70 < R2 < 0.90
Significant range of R2 (marked with red font): R2 > 0.90 Only one or two data point available, no reliable trend line could be drawn: N/A
It can be seen from table 46.6 that there is no correlation higher than 0.45. Nevertheless, appendix
It can be seen from table 46.6 that there is no correlation higher than 0.45. Nevertheless, appendix