Statistical Process Control Lab

by

Jenna Hartmann

Submitted to

Dr. C. G. Willson

ChE 253M Department of Chemical Engineering The University of Texas at Austin

Fall 2007

Statistical Process Control

Abstract

2

Contents

Introduction 4

Methods 4

Results 6

Conclusions and Recommendations 15

Appendices Appendix A 17 Appendix B 19 Appendix C 21

References 22

List of Tables Table 1: Process Capability Index Values 14

List of Figures Figure 1: Process Flow Diagram 4 Figure 2: Pure Dye Pre-Mixer X-bar Chart, Estimation Method 6 Figure 3: Pure Dye Pre-Mixer R Chart 7 Figure 4: Pure Dye Post-Mixer X-bar Chart, Estimation Method 7 Figure 5: Pure Dye Post-Mixer R Chart 8 Figure 6: Mix Stream Pre-Mixer X-bar Chart, Estimation Method 9 Figure 7: Mix Stream Pre-Mixer R Chart 9 Figure 8: Pre-Mixer Distribution Histogram 10 Figure 9: Mix Stream Pre-Mixer X-bar Chart, Standard 11 Deviation Method Figure 10: Mix Stream Post-Mixer X-bar Chart, Estimation Method 12 Figure 11: Mix Stream Post-Mixer R Chart 12 Figure 12: Post-Mixer Distribution Histogram 13 Figure 13: Modified (plotted without outliers) Post-Mixer Histogram 13 Figure 14: Mix Stream Post-Mixer X-bar Chart, Standard 14 Deviation Method Figure 15: Pre- and Post-Mixer Raw Concentration Data 15

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Statistical Process Control

Introduction

The purpose of this lab was to determine whether a continuous mixing process was “in control” by analyzing control charts and quantifying the benefits of an in-line mixer in the process. Statistical process control is a frequently used analytical tool for quality improvement programs. In this lab, the quality of the product stream (green dye and water solution) was analyzed using control charts (X-bar for the operating process level and R for variability) which were created by (1) an estimation method in JMP and (2) a standard deviation method. The resulting control charts were then compared to make quality control suggestions for the operating system.

Methods

The experimental apparatus for the Statistical Process Control lab is shown below in Figure 1. The apparatus consists of a large 20L container that holds the 0.2 wt% dye solution. Connected to this container is the recycle pump that mixes the solution to a uniform concentration. A peristaltic pump is also connected to the outlet of the container. This pump generates the continuous flow rate of dye solution to the rest of the apparatus. The water used for the dye and water mixture enters from the top left of the apparatus (as shown in Fig 1) at a junction before entering the first spectrometer. The mixed solution travels past the pre-mixer spectrometer before entering the in-line mixer. After the mixer, there is a second (post-mixer) spectrometer. The mixed solution then enters into a large graduated cylinder before exiting the system to the drain (Mullet, 2007).

Figure 1: Process Flow Diagram

The two detectors in this lab are spectrophotometers, or spectrometers for short, but they have slightly different path lengths (the length that the detecting light passes through the fluid). The first spectrometer has a path length of 0.56 cm, and the second spectrometer has a path length of 0.54 cm. Each of the two detectors uses the same tungsten halogen lamp as a light

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source. The light is directed from the lamp to the detectors by means of a fiber optic cable. Once the light reaches the spectrometer, it enters the flow cell and is emitted through the flowing fluid to a detector on the other side. The spectrometer measures the intensity of light passing through the fluid. The negative logarithm of the ratio of the intensity of the light that exits the flow cell versus the intensity of light of some reference is the absorbance (shown as Equation 1).

 I  A = − log  (1)  I O 

As the weight percent of the solution changes, the absorbance changes. A larger weight percent of dye means a larger absorbent peak due to the presence of more dye, and hence the less light that is able to pass through it. The absorbance is only a function of the amount of dye in the solution because it was “zeroed out” by taking measurements with a stream of pure water. The measurements taken with the pure water stream are then used as the reference intensity for absorbance calculations. The relationship between the amount of absorbance and the concentration is known as Beer’s Law, shown in Equation 2 below:

A = εcl (2) where c is the concentration, l is the path length of the flow cell, A is the absorbance, and ε is the extinction coefficient. Beer’s Law shows that a larger concentration is directly proportional to a larger absorbance (Mullet, 2007). Before the dye solution could be mixed with the tap water, a calibration curve needed to be created using a series of settings for the rotameter. The rotameter flow rate was measured at settings of 26%, 36%, and 51% of the maximum flow rate. The flow rate of the peristaltic pump was also measured by use of a bucket and stopwatch method. Dye solution was sent through the system without mixing with tap water, and the absorbance was taken before and after the in-line mixer. Then, dye solution and tap water were mixed, and the absorbance was measured before and after the in-line mixer. All the intensity data collected by the computer were then used to calculate the absorbance, and hence concentration, by Beer’s Law. Finally, control charts were created using the estimation and standard deviation methods. Random fluctuations in a process are always present, so processes are studied to determine if non-random variations exist. Non-random variations mean that the process can be modified to obtain statistically better data because the problems are from “assignable causes.” Control charts are an excellent way to interpret the data to determine what types of fluctuations are present. When looking at a control chart, the process is said to be “in control” if the data is varying only by random fluctuations. The term “in control” only corresponds to the statistical data, not the specifications needed on the product. A system is said to be out of control when there are both random and non-random variations. The capability index incorporates specified quality limits and predicts the product quality that can be expected from a process (Mullet, 2007).

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Results

The experimental results confirmed that the in-line mixer provided benefits for the process, but not enough to be considered for further use (e.g. in a factory). The mixed stream’s pre-mixer concentration data followed a random but non-normal distribution, and the post-mixer data followed a normal but non-random distribution. For a system to truly be in control, the data must be both normal and random (Mullet, 2007). Since neither the pre- nor post-mixer data met both of these requirements, the system was determined to be out of control. The mixer reduced the variability of the process (R bar = 0.017 pre-mixer and R bar = 0.00069 post-mixer), but the post-mixer x-bar chart shows trends in the data and an out of control system. The mixer proved capable of producing a ± 0.002% product in the pure dye stream (Cp pre-mixer = 2.28 and Cp post-mixer = 1.82) but not in the mix stream (water and dye) experimental runs (Cp = 0.092 and 1.13 pre- and post-mixer, respectively). First, the experimenters calibrated the spectrometer by taking a dark reference measurement using a stream of pure dye. Figures 2 and 3 display the pre-mixer X-bar and R charts for the pure dye stream. Figure 2 exhibits the non-random behavior of the system prior to entering the mixer. The control chart shows that the dye concentration varies around the lower control limit at first, then follows an upward trend, and finally maintains steady fluctuations around the upper control limit. This trend may have resulted because the experimenters did not allow the process to reach steady state before taking data readings. Figure 3 shows that the concentration of the dye before the mixer maintained a wide range of variability. The process did not vary to a state of being out of control, but the R chart shows that the pre-mixer conditions result in a wide range of resultant product, in this case, a wide range of dye concentration.

0.1705

0.1703

0.1701 UCL=0.170113

0.1699 Avg=0.16987 0.1697 LCL=0.169627 0.1695

0.1693 DYE Pre-MixerDYE Concentration (wt%) 16 32 48 64 80 96 112 Sample

Figure 2: Pure Dye Pre-Mixer X-bar Chart, Estimation Method

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0.0011

0.0009 UCL=0.000889 0.0007

0.0005 Avg=0.000421 0.0003

0.0001 LCL=0 -0.0001 DYE Pre-MixerDYE Concentration (wt%) 16 32 48 64 80 96 112 Sample

Figure 3: Pure Dye Pre-Mixer R Chart

Figures 4 and 5 present the control charts for the post-mixer results of the pure dye stream. Again, Figure 4 shows the same upward trend in the X-bar chart as in the pre-mixer X- bar chart but to a lesser degree. In the case of the post-mixer, the process appears to operate under more control, with the exception of one outlying point. However, although the post-mixer results show the benefits of the mixer, the concentration still varies over a wide range of values, as shown by the R chart in Figure 5.

0.1668 UCL=0.166669 0.1666

0.1664

0.1662 Avg=0.166209

0.166

0.1658 LCL=0.165749 0.1656 DYE Post-MixerDYE Concentration (wt%) 16 32 48 64 80 96 112 Sample

Figure 4: Pure Dye Post-Mixer X-bar Chart, Estimation Method

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UCL=0.00169

0.001 Avg=0.00080

0.000 LCL=0.00000

DYE Post-MixerDYE Concentration (wt%) 16 32 48 64 80 96 112 Sample

Figure 5: Pure Dye Post-Mixer R Chart

After calibrating the spectrometers with the pure dye stream, the experimenters then took measurements of a mix stream, which consisted of a mixture of equal proportions of dye and pure water. Figures 6 and 7 present the concentration data taken before the mixer. The X-bar chart (Fig 6) shows that the system process before the mixer performed at a fairly stable level with an average concentration of 0.09 wt% (ideal value of 0.10 wt%). By visual inspection, the control chart shows that the process behaved randomly with each subgroup average (i.e. data point on the control chart) varying below and above the overall average with equal probability. The R chart (Fig 7) proves this quantitatively. Upon randomization of the data and recalculation of the subgroup R values, the overall R average value remained the same (R average = 0.017 for both non-randomized and randomized data; see Appendix B). Because the average R value does not change upon randomization, the experimental pre-mixer data prove to be a set of randomly occurring points with no trend lines or out of control points. Therefore, the pre-mixer process fulfilled one requirement for statistical control because any variations in the process resulted from chance variations in the process and not because of some other controllable aspect of the system.

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0.105 UCL=0.10403

0.100

0.095 Avg=0.09416

0.090

0.085 LCL=0.08429 MIX Pre-MixerMIX Concentration (wt%) 16 32 48 64 80 96 112 144 176 208 Sample

Figure 6: Mix Stream Pre-Mixer X-bar Chart, Estimation Method

0.040 UCL=0.03618 0.030

0.020 Avg=0.01711

0.010

0.000 LCL=0.00000 MIX Pre-MixerMIX Concentration (wt%) 16 32 48 64 80 96 112 144 176 208 Sample

Figure 7: Mix Stream Pre-Mixer R Chart

Although the pre-mixer data showed statistical control through randomization, this data did not follow a normal distribution, the second requirement for a system that is truly in control (see Fig 8). The pre-mixer data does not distribute evenly in a standard and normal Gaussian curve, but displays a more skewed distribution because it is not well mixed. Concentrations from sample to sample deviate more from the average than a well-mixed solution, so the pre-mixer conditions introduce much more variability, resulting in a broader range of data points and a flattened distribution curve.

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Pre-Mixer Distribution Histogram

120 100 80 60 40 Frequency 20 0 0.078 0.080 0.082 0.084 0.086 0.088 0.090 0.092 0.094 0.096 0.098 0.100 0.102 0.104 0.106 0.108 0.110 Other Sample (Concentration wt%)

Figure 8: Pre-Mixer Distribution Histogram

In comparison to the estimation method used to create the control charts above, a standard deviation method also allows for process control analysis. Figure 9 shows the control chart produced by the standard deviation method. The chart has an average x-bar value of 0.094, upper control limit of 0.12, and a lower control limit of 0.072. The x-bar average matches exactly to that predicted by the estimation method and the UCL and LCL compare within 15% and 10% difference, respectively.

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Pre-Mixer X-bar Chart (Standard Deviation Method)

0.14

0.13

0.12

0.11

0.1

0.09

0.08

Concentration (wt%) Concentration 0.07

X-bar Values 0.06 UCL LCL 0.05 Average X-bar 0.04 0 200 400 600 800 1000 1200 Sample Number

Figure 9: Mix Stream Pre-Mixer X-bar Chart, Standard Deviation Method

Analysis of the post-mixer data in comparison to the pre-mixer data allowed for investigation into the pros and/or cons of the in-line mixer. Below, Figures 10 and 11 present the X-bar and R charts of the flow system after the mixer. Ideally, the average x-bar of the post- and pre-mixer are equivalent and, in this experiment, are within 1% difference. The X-bar chart, like the pure dye pre- and post-mixer X-bar charts, shows the same upward trend tendency at around the same time (approximately sample number 64). The mixed stream control chart also shows a downward trend at a later time, which can be speculated to steady off into a seemingly controlled process. The apparent upward and downward trend could be due to data acquisition during the process startup. The curve seen in the X-bar chart could be due to dye that accumulated in the mixer during startup and then released in one pulse, yielding higher concentrations of dye for a range of time before returning to a normal operating level. The process appears to level off to a steadier state, albeit the lower control limit in this control chart. The R chart (Fig 10) for the post-mixer process shows a constriction of the upper and lower control limits compared to the pre-mixer R chart. This shows that the post-mixer results experience less variability in the resulting product. Although the post-mixer has less variability, the collected data did not exhibit randomness. After randomizing the data, the average R value changed from 0.00069 for the non-randomized data to 0.0012 for the randomized data (see Appendix B). For truly randomized data, the average R value does not change with randomization. Therefore, although the post-mixer process appears relatively in control, it is not by statistical means. The two extreme outlying points in the X-bar and R charts could be due to dirt present in the system, which may have given faulty data.

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0.0965 0.096 0.0955 0.095 0.0945 0.094 0.0935 UCL=0.093505 0.093 Avg=0.093109 0.0925 LCL=0.092713 0.092 MIX Post-MixerMIX Concentration (wt%) 16 32 48 64 80 96 112 144 176 208 Sample

Figure 10: Mix Stream Post-Mixer X-bar Chart, Estimation Method

0.013

0.011

0.009

0.007 0.005

0.003 UCL=0.00145 0.001 Avg=0.00069 -0.001 LCL=0.00000 MIX Post-MixerMIX Concentration (wt%) 16 32 48 64 80 96 112 144 176 208 Sample

Figure 11: Mix Stream Post-Mixer R Chart

Although the post-mixer data did not express randomness, it did exhibit a normal distribution, which is a requirement of a statistically in control process. Figure 12 displays the histogram with all of the data points included. Removal of the outlying data points makes the histogram appear normal, as shown in the modified histogram of Fig 13. The normal distribution curve confirms that approximately 99% of all data points taken lie within the control limits, although the above control chart shows otherwise. The fact that the above X-bar chart (Fig 10) does vary outside the control limits while the normal distribution curve displays the contrary proves that the post-mixer is more sensitive to non-random fluctuations in the process than the pre-mixer. This sensitivity is good for the system because non-random fluctuations in the process can be easily detected.

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Post-Mixer Distribution Histogram

350 300 250 200 150

Frequency 100 50 0

Other 0.09180.09250.09330.09400.09480.09550.09630.09700.09780.09850.09930.10010.10080.10160.10230.1031 Sample (Concentration wt%)

Figure 12: Post-Mixer Distribution Histogram

Modified Post-Mixer Distribution Histogram

350 300 250 200 150

Frequency 100 50 0 0.0918 0.0922 0.0925 0.0929 0.0933 0.0937 0.0940 0.0944 0.0948 Sample (Concentration wt%)

Figure 13: Modified (plotted without outliers) Post-Mixer Histogram

Figure 14 presents the X-bar chart for the standard deviation method. In comparison to the estimation method, both charts follow the same curvature. In comparing the standard deviation method with that of the estimation method, the x-bar average values are equal, and the UCL and LCL compare within 1.6% and 1.8% difference, respectively. The two methods for the post-mixer data compare much better than those for the pre-mixer data. This occurs because the control limits are tighter for the post-mixer and approximation values have less room for error.

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Post-Mixer X-bar Chart (Standard Deviation Method)

0.096 X-bar Values UCL 0.0955 LCL Average X-bar 0.095

0.0945

0.094

0.0935

0.093 Concentration(wt%)

0.0925

0.092

0.0915

0.091 0 200 400 600 800 1000 1200 Sample Number Figure 14: Mix Stream Post-Mixer X-bar Chart, Standard Deviation Method

Analysis of the in-line mixer benefits concludes that the in-line mixer, in this lab, did not prove beneficial to the system. The mixer proved beneficial in the pure dye experimental runs but not in the mix stream (water and dye) experimental runs (see Table 1). A process capability index (Cp) value of 1.3 is an acceptable value and higher values signify a more capable process (Mullet, 2007). A higher Cp value means that there are tighter control limits and less deviation from output to output which resembles a process more capable of producing the specified product. The pure dye shows higher Cp values than the mix stream, but the Cp value decreased with the addition of the mixer. The mixed stream system showed benefits with the addition of the in-line mixer (Cp increased), but not enough to consider the process capable of a ± 0.002% product specification limit.

Table 1: Process Capability Index Values Pre-Mixer Post-Mixer

Pure Dye 2.28 1.82

Mix Stream 0.092 1.13

Figure 15 below displays the “raw” concentration data collected for both the pre- and post-mixer. The post-mixer yields concentrations that vary more closely to a concentration of 0.093 (post-mixer x-bar value) whereas the pre-mixer data vary more widely around 0.094 (pre- mixer x-bar value). Not including the two outlying data points, the post-mixer data shows that the process benefited from the in-line mixer. The product stream post-mixer maintained a more

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constant value, whereas the system without the mixer (i.e. pre-mixer) varied more wildly and the probability of producing the same product over time decreased. The average R value for the pre- mixer (0.017) differed from the post-mixer average R value (0.00069) by approximately 24%. The average R value shows that the variability in the pre-mixer was about 24% greater than the average R value for the post-mixer. This was to be expected because the pre-mixer data did not undergo even mixing (turbulence signifies better mixing because of its tumultuous flow behavior). Because the flow regime is laminar, there existed greater variability in the pre-mixer data. Increased flow rate would help to increase the uniform mixing of the process and more uniform mixing leads to a more in control process and less variability.

Pre- and Post-Mixer "Raw" Concentration Data

0.12

0.11

0.1

0.09 Concentration(%wt) 0.08

0.07 Pre-Mixer Concentration

Post-Mixer Concentration

0.06 0 200 400 600 800 1000 1200 1400 1600 Time (sec) Figure 15: Pre- and Post-Mixer Raw Concentration Data

Conclusions and Recommendations

The experimental results confirm that the in-line mixer, though beneficial to the process, did not perform well enough to render the system capable of producing product at ± 0.002% specification limits (Cp = 1.13, where Cp = 1.3 is the acceptable limit). The mixer reduces the variability in the final product, but the data followed non-random trends in the post mixer X-bar control chart, which means that not all fluctuations relied solely on chance. The mixer also provided a normal distribution of data for the product stream, but because the data maintained non-random tendencies, the mixer did not perform in statistical control. Therefore, the

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experimenters do not recommend that the mixer be used in factory settings if the aim is to produce a product within specification limits. Autocorrelation occurs if samples are collected too quickly after one another. Although the mix stream’s pre-mixer data has non-random data and follows an upward and downward trend, autocorrelation did not cause this. Calculation of the velocity in the ¼” tubing yielded a value of 8.15 cm/s (0.2 mph) and a Reynolds number of 752 which signifies that the fluid flows in the laminar regime. Data collection occurred every second and with a velocity of 8.15 cm/s, autocorrelation did not cause the trends found in the data. Use of a spectrometer in this experiment yielded very good data as it is one of the best and only methods of detecting concentration in a flowing stream. The spectrometer responded to small variations in output concentrations without being too sensitive. However, because the stream in this experiment involved laminar flow, the spectrometer also had its disadvantages. In laminar flow, the parabolic velocity profile dictates a concentration profile within the flowing stream. When the spectrometer is set to pass directly through the most concentrated laminar region of flow at all times, then any variation in the flow, and hence the disruption of the concentrated region, gives increased variability in the resulting data. As concentration stratifies radially from the center to the walls of the tubing, any misplacement or adjustment of the spectrometer would yield inaccurate readings, especially if the pre- and post-mixer spectrometers are not adjusted equally. The experimenters advise being cautious around the apparatus and avoid adjusting the spectrometers without proper instruction. In addition, the experimenters suggest that irradiative detecting instruments could also be employed as a means of measuring concentration; however, these methods may be more expensive, and they would require that the green dye be marked with radioactive elements, increasing safety hazards in the lab. One cause for non-randomness in the post-mixer data could be due to startup reasons or a malfunction in the system. If the pump does not perform properly, then the pulse seen in the post-mixer data could result. One way to make the process in control, or rule out any non- random influence, is to wait for the process to reach steady state. In order to rule out the pulse seen in the post-mixer data, the experimenters recommend that the experiment be run so that the mixture flows through the apparatus for about five minutes before data acquisition. Also, the mixing response to turbulent flow is unknown, so further experimentation with turbulent regime flow would be required to test the mixer performance.

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Appendix A

Appendix A displays the sample calculations for calculated values presented in the lab report.

Pre-Mixer Flow Cell Path Length

 I  A = − log  (A-1)  I O 

where A = absorbance I = transmitted light, average value from pure dye data IO = reference light intensity, average value from pure water data

 232.11  A = −log   3000.62  (A-2) A = 1.11

A = εcl (A-3)

where ε = extinction coefficient = 12.5 (from graph in lab manual) c = concentration l = path length

c = A lε 1.11 c = 12.5⋅ 0.535

c = 0.166 %wt

Reynolds Number:

ρ ⋅V ⋅ D Re = (A-4) µ

where ρ = density, g/cm³ μ = viscosity, g/cm s V = velocity, cm/s D = diameter, cm (Schedule 40 ¼” steel pipe = 0.91 cm)

(1)(8.15)(0.91) Re = 0.01 Re = 742

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***Flow is in the laminar regime

Upper/Lower Control Limits

UCL = X + A2 R (A-5)

where X-bar-bar = average x-bar A2 = 0.58 from lab manual appendix R-bar = average ranges from each subgroup

UCL = 0.094 + 0.58(0.017)

UCL = 0.104

Process Capability

Cp = (USL – LSL)/6s = (USL – LSL)/(UCL-LCL) (A-6) Cp = (0.096-0.092)/(0.12-0.07) Cp = 0.09

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Appendix B

Appendix B presents any graphs and/or data tables relevant to the lab report.

Rotameter Calibration Curve

60

50

40

30 y = 12.413x + 2.239 2 20 R = 0.995

Rotameter Setting 10

0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 Flow Rate (mL/s)

Figure B-1: Rotameter Calibration Curve

0.110

0.105 UCL=0.10389 0.100

0.095 Avg=0.09416 0.090

Mean ofMean Random Pre-Mix 0.085 LCL=0.08443

16 32 48 64 80 96 112 144 176 208 Sample

Figure B-2: X-bar Chart of Randomized Pre-Mixer Data

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UCL=0.03566 0.030

0.020 Avg=0.01686

0.010 Range of Random Pre-Mix 0.000 LCL=0.00000

16 32 48 64 80 96 112 144 176 208 Sample

Figure B-3: R Chart of Randomized Pre-Mixer Data

0.0955

0.095

0.0945

0.094 UCL=0.093771 0.0935

0.093 Avg=0.093108 Mean ofMean Random Post-Mix 0.0925 LCL=0.092446 16 32 48 64 80 96 112 144 176 208 Sample

Figure B-3: X-bar Chart of Randomized Post-Mixer Data

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0.013

0.011

0.009

0.007 0.005

0.003 UCL=0.00243 0.001 Avg=0.00115 Range of Random Post-Mix LCL=0.00000 -0.001 16 32 48 64 80 96 112 144 176 208 Sample

Figure B-4: R Chart of Randomized Post-Mixer Data

Appendix C

Appendix C presents discussion of details not included in the lab report and possible safety hazards in the lab.

The green dye solution used in the lab absorbs visible light wavelengths at two different absorption peaks (graph in lab manual and not attached here). This occurs because any color that humans see is the absorbance of every color except the color seen. For example, with the color green, every color in the visible light spectrum is absorbed except the green spectrum. The visible light spectrum covers the range of 400 to 700 nm. The wavelength for blue is about 475 and about 570 for yellow. Green falls at a wavelength of about 510 nm. Because green is nearly in the center of the visible light spectrum, there are two curves present on the absorbance curve. One curve is the absorbance of the colors before it (violet and blue) and the other curve are the colors absorbed after it (orange, yellow, red). Anything outside of the 400-700 nm range is ultraviolet light and cannot be seen by humans (Kusterer, 2007). Safety hazards associated with this lab include slipping on spilled water and possible electric shock from water spilling on any of the electronic devices. These safety issues can be prevented by not spilling water on the floor when transporting it and by being careful. There is a piece of tubing connecting the graduated cylinder to the drain. If all of the water does not go into the drain, then it should be cleaned immediately to prevent a safety hazard in the lab. The dye used in the lab is not toxic, but care should be used when pouring the dye to ensure it does not get on the clothes or in the eyes of anyone in lab.

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References

Kusterer, John M. “What wavelength goes with a color?” Atmospheric Science Data Center. 28

Sept. 2007. Nasa. 2 Dec. 2007

Wavelengths_for_Colors.html>.

Mullet, Ian. Statistical Process Control Lab. Austin: n.p., 2007. This lab manual served as the

reference for equations and theoretical information about the laboratory experiment and

technique.

“Steel Pipes Dimensions - ANSI Schedule 40.” The Engineering Toolbox. 2005. 2 Dec. 2007

.

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