Statistical Process Control: Process and Quality Views

Table 1: n-hexane specific gravity data

    Statistical Process Control (SPC) provides a way to monitor chemical and other   processes.   We'll focus on continuous chemical processes and how the process and quality control departments utilize SPC.  Process control engineers use SPC to monitor a process's stability, consistency and overall performance.  Quality control engineers use SPC to see if the process is functioning within quality standards.  In industry, these two departments work together to monitor a chemical process.  SPC, in a classical sense, will not reveal much about the quality of the product.  For example, a process may be operating very well and in a very stable manner...as far as the process engineer is concerned, everything is fine.  However, if the process is currently 20% below the quality standard for A or top grade material, it would be difficult to say that the process is fine.
    To help introduce the basics of SPC, we'll assume that the variable being monitored is the specific gravity (SG) of n-hexane as it is being produced.   We'll assume that the SG is measured four times per day at 0300, 0900, 1500, and 2100 by the plant's laboratory.  Table 1 shows the results for a three day time period.  A-grade industrial n-hexane must have a SG between 0.61 and 0.69.   First, we'll see how the process engineer analyzes this data.

SPC Overview:
   In a continuous chemical process, two types of charts are commonly used: individual value or X-bar charts and moving range (MR) or R-bar charts.  X-bar charts are used on a regular basis to monitor the process during a time of change.  For example, an R-bar chart would be appropriate if you were changing the feeds to the process.  The R-bar chart weights more recent data more heavily than historical data.
    The chemical industry typically uses one of two types of process control.  3-sigma control specifies quality limits nearly equal to process limits.   6-sigma control specifies quality limits that are twice as large as control limits.   We'll focus on the 3-sigma system.
    With all of the different types of limits, it's easy to become confused.  For our n-hexane process, we'll have 6 different limits we'll consider.   Three UCL's (Upper Control Limits) and three LCL's (Lower Control Limits).

UCL (calculated) = statistical upper control limit
UCL (process) = pre-determined, acceptable process upper control limit
UCL (quality) = pre-determined, acceptable quality upper control limit

LCL (calculated) = statistical lower control limit
LCL (process) = pre-determined, acceptable process lower control limit
LCL (quality) = pre-determined, acceptable quality lower control limit

The process limits are those which define boundaries of operation for the process or an acceptable operating value.  The quality control limits are those used to "grade" material.  The term "quality limits" will refer to the A grade or top grade material limits.  You should realize that there are also B and C grades of materials that companies often sell as well.  The limits of these other grades vary accordingly.  Essentially, the farther away from specifications a product is, the lower the grade, and its value decreases sharply.
    Typically in a 3-sigma system, the process limits are said to be "tighter" than the quality limits by 5-10%.  This is done so that even if the process exceeds process limits by a small amount, it will still be within quality standards.  However, the 6-sigma system dictates that the process limits be half of the quality limits.  For example, if you had an upper quality control limit of 100, the upper process control limit in a 6-sigma system would be 50 while a 3-sigma system may have an upper process control limit of around 90.  Basically, a 6-sigma system requires more strict (and sometimes unrealistic) control, depending on the process.   This is why many chemical manufactures implement the 3-sigma system.  Now that we've discussed the different types of limits and charts involved, let's see how our system is performing!

SPC: X-bar charts
    Start by calculating the average for the data points:
                                                       (1)
where Xi is each individual result and n is the total number of results.

Now,
        (2)
           
            (3)
       
            (4)

            (5)


For our system:

In a 3-sigma system, Z is equal to 3 [Equations 2 and 3] (hence its name) and a 6-sigma system uses Z=6, therefore:
UCL (calculated) = 0.65 + 3(0.0056) = 0.67
LCL (calculated) = 0.65 - 3(0.0056) = 0.63

    As mentioned before the other control limits are set depending on the quality of the product needed.  A-grade n-hexane must be between 0.61 and 0.69 (these are the quality limits).  Typical process limits may then be 0.62 to 0.68.  Now we know all of the limits for our current data:
UCL (calculated) = 0.67
UCL (process) = 0.68
UCL (quality) = 0.69

LCL (calculated) = 0.63
LCL (process) = 0.62
LCL (quality) = 0.61

    At this point, it's tempting to conclude that since the calculated limits are inside the process and quality limits, the process is operating perfectly.   But let's have a look at the X-bar chart:

    Figure 1 shows the performance from 2/25/99 to 2/28/99.   According to the definition of "in control", the process should meet four criteria:
1.  No sample points outside of process limits
2.  Most points near average
3.  Nearly equal number of points above and below average
4.  Points are randomly distributed
According to Figure1, only conditions 3 and 4 are being met.  This process should be examined for process upsets or interruptions in stability.  After the appropriate process changes were made, another X-bar chart was constructed over another 4 day period, Figure 2 below shows these results:

    After the process improvements, the data suggests that the process is in control and all four criteria for control are being met.  Figure 2 shows how you should aim to control your process.

SPC: R-bar charts
    R-bar charts utilize chart factors that are typically found in statistical references.  Table 2 shows a portion of such a chart for 3-sigma control:

Table 2: 3-sigma Control Chart Factors

    The UCL (calculated) are LCL (calculated) are defined by:
                                    (6)

                                        (7)

                                        (8)

where MR (moving range) is the absolute value of the difference between the current data point and the preceding data point.  The number of sub-groups is an area that most people do not agree upon.  For example, if you group results by week and you're analyzing data for a month you could use 4 sub-groups.  As a general rule, if your continuous process has been operating under the same specifications over the time of your analysis, you may assume 2 sub-groups.  This is the approach we'll use for our system.
    Let's assume that our plant also produces glycol which has an average specific gravity of 1.11.  An R-bar chart provides an effective means of monitoring the transition from n-hexane (SG=0.65) to glycol (SG=1.11).  Monitoring the individual results (X-bar chart) in conjunction with the R-bar chart will paint a very clear picture of the transition.  The data in Table 3 shows the 4-day transition.   The feeds were changed just prior to this data being recorded.  What we're seeing is the n-hexane leaving the system and the glycol showing up gradually.

Table 3: n-hexane to glycol transition data

    Now, to form the R-bar chart, we graph the data/time versus the moving range.  We can assume 2 sub-groups (n-hexane and glycol over a short period of time).  The moving range for 3/15/99 at 0900 was calculated by |(0.63-0.65)|=0.02.  Typically, there will be no process or quality control limits for R-bar charts.  For this transition:

UCL(calculated) = (3.27)(0.043) = 0.14
LCL(calculated) = (0)(0.043) = 0.00

Figure 3 shows the R-bar chart for the transition from n-hexane to glycol.  While using an X-bar chart alone during times of change in a system is feasible, it is sometimes difficult to graph the data due to the potentially large differences in results.   Another consideration is that in data compiling, a R-bar chart appearing between X-bar charts is a nice way to show a transition phase has occurred.  R-bar charts are also useful when plotting data over a large time span.  To show the contrast in the two types of charts, the x-bar chart during the transition is shown in Figure 4.

    The arrow in Figure 4 shows the point at which the SG of the glycol has entered the quality control range.  This is a very important point because the system output must be directed to a glycol storage tank at this point (rather than the waste container used during the transition).  This is why the charts should be used in conjunction with one another, and an R-bar chart should not be used alone during a transition.

Uses of R-bar charts:
1.  Keep a record of when process changes or feed changes occurred
2.  Record of how long the process took to stabilize
3.  Show long history of a process or piece of equipment

    If or whenever you use R-bar charts, remember that they tell you nothing about the actual value of the results, only deviations from one result to the next.

SPC: Danger signs and where to start looking
    As a process or quality engineer, you'll eventually come across some charts that make your eyes pop out and spell "O-V-E-R-T-I-M-E".   Imagine going to work one morning and find the chart below:

    Depending on where the results came from, the problem could be several things.
If the results are from online measuring devices:
1.  Check the operator's log for any abnormal behavior during the time that the results were out of standard.
2.  Check the calibration schedule for the measuring device.

If the results are from a laboratory:
1.  Check laboratory equipment for correct calibration.
2.  Review laboratory notes on the tests for any errors that may have been made in the testing procedure.
3.  Ask the laboratory technician if he/she remembers anything strange about the tests.  For example, sample collection container abnormalities that may have led to contamination.

Any investigating beyond these ideas may begin to be counterproductive.  However, the process should be monitored closely to see if this result is repeated later.  The process has returned to normal operation and the out of standard results were not a serious compromise of quality.

    Now suppose you find a chart resembling Figure 6:

You may initially think that since no results are out of the quality control standards that you don't have a problem.  To the contrary, unless there has been an intentional process change, you have a very serious problem.  It's not a matter of where the process has been or where it is now, but where it is going.   This type of trend cannot be attributed to simple error, there is something seriously wrong!  Depending on many factors, you must find a place to start investigating.  You may want to start with equipment that can immediately affect the SG.  Look closely at the process data over the past few days.  It can be helpful to compare data over a comparable period of time when the process was in control versus this new trend that you're seeing.  I might suggest starting with the separation equipment.

    Until this point we've considered only one characteristic of n-hexane, specific gravity.  Process engineers must simultaneously monitor all important characteristics of a product.  Suppose your manager presents you with the following two charts:

You notice that the SG seems to be fine while the concentration has dropped off dramatically.  While SG is generally a good indicator of concentration, it doesn't appear to be so in this situation.  Since you have no reason to doubt the accuracy of these results at this time, where should you start?  The first question you should answer is: "What else is in the stream that is lowering the concentration?"  A quick look at the gas chromatograph show that on 3/4/99 at 1500 (when the n-hexane concentration was at 92%) there was also significant amounts of two other chemicals: 2-methyl-butene-3 (SG=0.63) and isoprene (SG=0.68)[hypothetical components].  Since the 2-methyl-butene-3 is lighter than n-hexane and isoprene is heavier than n-hexane, the contaminant mixture did not force the specific gravity out of standard, but the concentration is being seriously affected (this is why you monitor both SG and concentration).  Now all you have to do is find out how it got there.  I might start with heat exchangers that may be leaking.