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Statistical Process Control Lab

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					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




                                           3
                                 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

                                              4
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 
                                                 I    
                                                                                     (1)
                                                  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  cl                                (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).




                                              5
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.
          DYE Pre-Mixer Concentration (wt%)




                                              0.1705

                                              0.1703

                                              0.1701                                                UCL=0.170113

                                              0.1699
                                                                                                    Avg=0.16987
                                              0.1697
                                                                                                    LCL=0.169627
                                              0.1695

                                              0.1693
                                                          16     32    48       64   80    96    112
                                                                        Sample

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




                                                                            6
         DYE Pre-Mixer Concentration (wt%)
                                                    0.0011

                                                    0.0009                                                    UCL=0.000889
                                                    0.0007

                                                    0.0005
                                                                                                              Avg=0.000421
                                                    0.0003

                                                    0.0001
                                                                                                              LCL=0
                                               -0.0001
                                                                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.
               DYE Post-Mixer Concentration (wt%)




                                                    0.1668
                                                                                                             UCL=0.166669
                                                    0.1666

                                                    0.1664

                                                    0.1662                                                   Avg=0.166209

                                                     0.166

                                                    0.1658
                                                                                                             LCL=0.165749
                                                    0.1656
                                                                16       32    48        64   80   96       112
                                                                                Sample

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




                                                                                     7
          DYE Post-Mixer Concentration (wt%)
                                                                                                 UCL=0.00169



                                               0.001
                                                                                                 Avg=0.00080




                                               0.000                                             LCL=0.00000

                                                       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.




                                                                        8
           MIX Pre-Mixer Concentration (wt%)
                                               0.105
                                                                                                    UCL=0.10403

                                               0.100

                                               0.095
                                                                                                    Avg=0.09416

                                               0.090

                                               0.085
                                                                                                    LCL=0.08429

                                                       16 32 48 64 80 96 112     144   176   208
                                                                        Sample

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

                                               0.040
           MIX Pre-Mixer Concentration (wt%)




                                                                                                    UCL=0.03618
                                               0.030


                                               0.020
                                                                                                    Avg=0.01711

                                               0.010


                                               0.000                                                LCL=0.00000
                                                       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.




                                                                          9
                                                   Pre-Mixer Distribution Histogram

                     120
                     100
         Frequency


                     80
                     60
                     40
                     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.




                                                                            10
                                                 Pre-Mixer X-bar Chart (Standard Deviation Method)


                             0.14

                             0.13

                             0.12

                             0.11
       Concentration (wt%)




                              0.1

                             0.09

                             0.08

                             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.




                                                                11
          MIX Post-Mixer Concentration (wt%)
                                               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
                                                        16 32 48 64 80 96 112     144   176   208
                                                                         Sample

                                                Figure 10: Mix Stream Post-Mixer X-bar Chart, Estimation Method
          MIX Post-Mixer Concentration (wt%)




                                                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
                                                        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.


                                                                          12
                                           Post-Mixer Distribution Histogram

                        350
                        300
                        250
        Frequency




                        200
                        150
                        100
                         50
                             0




                              er
                        0. 8

                        0. 5

                        0. 3

                        0. 0

                        0. 8

                        0. 5

                        0. 3

                        0. 0

                        0. 8

                        0. 5

                        0. 3

                        0. 1

                        0. 8

                        0. 6

                        0. 3
                            31
                            1

                            2

                            3

                            4

                            4

                            5

                            6

                            7

                            7

                            8

                            9

                            0

                            0

                            1

                            2


                           th
                          09

                          09

                          09

                          09

                          09

                          09

                          09

                          09

                          09

                          09

                          09

                          10

                          10

                          10

                          10

                          10
                         O
                        0.




                                                    Sample (Concentration wt%)

                                       Figure 12: Post-Mixer Distribution Histogram



                                       Modified Post-Mixer Distribution Histogram

                        350
                        300
                        250
            Frequency




                        200
                        150
                        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.




                                                        13
                                                   Post-Mixer X-bar Chart (Standard Deviation Method)


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


                              0.0945
        Concentration (wt%)




                               0.094


                              0.0935


                               0.093


                              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

                                                                     14
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
   Concentration (%wt)




                         0.09




                         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


                                                              15
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.




                                            16
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 
                                                I    
                                                                                    (A-1)
                                                 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  cl                                        (A-3)

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

                                     cA
                                          l
                                            1.11
                                     c
                                        12.5  0.535

                                     c  0.166 % wt


Reynolds Number:

                                              V  D
                                      Re                                            (A-4)
                                                

       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

                                             17
***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




                                        18
Appendix B

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


                                                                               Rotameter Calibration Curve


                                            60

                                            50
         Rotameter Setting




                                            40

                                            30                                                                      y = 12.413x + 2.239
                                            20                                                                           R2 = 0.995

                                            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
                             Mean of Random Pre-Mix




                                                      0.105
                                                                                                                         UCL=0.10389
                                                      0.100

                                                      0.095
                                                                                                                         Avg=0.09416
                                                      0.090

                                                      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




                                                                                          19
                                                                                              UCL=0.03566


             Range of Random Pre-Mix
                                       0.030


                                       0.020
                                                                                              Avg=0.01686

                                       0.010


                                       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
Mean of Random Post-Mix




                                       0.095

                          0.0945

                                       0.094
                                                                                              UCL=0.093771
                          0.0935

                                       0.093                                                  Avg=0.093108

                          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




                                                                    20
                                     0.013


          Range of Random Post-Mix
                                     0.011

                                     0.009

                                     0.007
                                     0.005

                                     0.003
                                                                                           UCL=0.00243
                                     0.001                                                 Avg=0.00115
                                                                                           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.




                                                                  21
References


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

       Sept. 2007. Nasa. 2 Dec. 2007 <http://eosweb.larc.nasa.gov/EDDOCS/

       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

       <http://www.engineeringtoolbox.com/orifice-nozzle-venturi-d_590.html>.




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