Example 3

A simple out-of-control example with a sample constructed control chart.

You have been analyzing the odd operation of a temperature sensor in one of the plant's CSTR reactors. This particular CSTR's temperature sensor consists of three small thermocouples spaced around the reactor: T1, T2, and T3. The CSTR is jacketed and cooled with industrial water. The reaction taking place in the reactor is moderately exothermic. You know the thermocouples are working fine; you just tested them, but a technician suggests the CSTR has been operating out of control for the last 10 days. There have been daily samples taken and there is a control chart created from the CSTR's grand average and standard deviation from the year's operation.

You are assigned to see if the CSTR is operating out of control. The grand average is 307.47 units of temperature and the grand standard deviation is 4.67 units of temperature. The data is provided for construction of the control chart in Table 1 and the data from the last 10 troublesome days is shown in Table 2. You decide to plot the troublesome data onto the control chart to see if it violates any stability rules.

value calculate
Upper Control Limit 316.60 μ + R3σ
Upper Zone A Threshold 313.56 μ + (2/3)* R3σ
Upper Zone B Threshold 310.51 μ + (1/3)* R3σ
Grand Average Value 307.47 μ
Lower Zone B Threshold 304.42 μ - (1/3)* R3σ
Lower Zone A Threshold 301.38 μ - (2/3)* R3σ
Lower Control Limit 298.33 μ - R3σ

μ 307.47
σ 4.67
R3 1.954

Table 3-1. Data for Construction of Control Chart

The way I found A_3 or in this case, R_3, I used the control charts constants table which is found on this wiki page. I decided to use the x-bar using the standard deviations) but you can also use use the range). I found that the value for n(number of subgroups) is three since the CSTR's temperature sensor consists of three small thermocouples (T1,T2,T3). Therefore by looking at the constant chart, I get A_3( or R_3 in this case) to be 1.954. Here's the table below:

Subgroup x-bar chart
Using Ra
 Using Sa
n
 A2
 A3
2 1.886 2.659
3 1.023 1.954
4 0.729 1.628
5 0.577 1.427
6 0.483 1.287

Also, you will notice if you used the range instead of the standard deviation to determine the UCL,LCL, etc. that the values will be roughly the same. Here's the table in comparing the values of UCL and LCL using either A_2 (range) or A_3(stdev):

A3 (stdev)
 A2 (range)
1.954 1.023
Using Sdev
 Using Range
UCL 316.5997379 316.473784
LCL 298.3347317 298.460686

Note: These values were using the same grand average (307.47), the grand standard deviation (4.67) and the grand range (8.80)

Sample Day
 T1 T2 T3
1 299.82 310.26 306.60
2 311.68 307.04 300.90
3 310.94 311.68 306.83
4 325.00 308.82 304.97
5 321.43 300.98 311.23
6 320.74 308.97 305.26
7 314.77 313.25 303.36
8 332.75 302.76 306.11
9 319.87 296.81 305.95
10 315.21 314.99 309.60

Table 3-2. Sample Data from Past 10 Troublesome Days

Solution

When the sample data was graphed onto the control chart, the image below was seen.


Figure 3-1. 10-Day Data Graphed Onto Control Chart

We can see from the control chart that the CSTR system is clearly out of control. Each thermocouple was tested to see which stability rules it violates.

The first thermocouple (T1) violates every stability rule.

  • Rule 1 - Several points from the T1 data fall above the upper control line.
  • Rule 2 - There are many instances where at least two out of three consecutive points fall above the zone AB threshold.
  • Rule 3 - There are eight consecutive points falling above the BC threshold.
  • Rule 4 - Nine consecutive points fall above the mean value.

Judging on this thermocouple's performance, we can say that the system is out of control, but we will analyze the other thermocouples' performance for good measure.

The second thermocouple (T2) violates stability rule 1, 2, and 3.

  • Rule 1 - One point falls below the lower control line.
  • Rule 2 - Two consecutive points (samples 9 and 10) fall beyond the AB threshold.
  • Rule 3 - Of the last five samples from T2, four are beyond the BC threshold.

The third thermocouple (T3) does not violate any stability rules and the results it displays are within control.

This system is out of control because the data from the thermocouples falls beyond the threshold rules for the unit's control chart. This could be explained with many potential situations. One is explained below.

If the CSTR's agitator is knocked loose, the agitation could become erratic. The erratic agitation could create eddy currents and hot spots in the CSTR.

The entire system is out of control because you know that the thermocouples are operating fine and more than one thermocouple violates the stability rules.