Design Of Experiment(DOE)

I was tasked to do Case Study 2 in this activity of DOE.

Case Study 2 Scenario is shown below:

In a wastewater treatment facility, a combination of coagulant chemicals, treatment temperature,  and stirring speed were identified as critical factors to treat the wastewater to produce clean water. The clean water produced is recycled back into the main process and at the same time reduce the amount of pollutant discharged by the plant. 8 runs were performed and the data are shown below. 

The response variable (y) is the amount of pollutant discharged (lb/day) 

A = concentration of coagulant added, 1% and 2% by weight 

B = treatment temperature, 72°F and 100°F

C = Stirring speed, 200 rpm and 400 rpm

8 runs were performed and below is the data given.

Full Factorial  Data Analysis

The 3 factors have 2 different levels which represent the lower level and higher level. The low and high values for each factor indicate the "-" and "+' signs respectively. The y value is typed in according to the factor levels as this case only has one replicate, we will only have to type in one column which is R1. Then, the average will be computed. Next, the reference be should be filled up so that we know which factors are used for respective alphabets. This eases our understanding to determine the factors. 

The excel sheet will auto calculate the values for us to determine the significance of the effects when it is positive and negative. The picture below shows the average value for each factor depending on the "-" and "+" level of factors. 

Then, plot a line graph of the discharge flow rate of pollutants vs the factor levels using the average values that we have collected from each factor of high and low levels.

Ranking of Factors from Most Significant to Least Significant: C>A>B

Ranking	Factors
1	C (Stirrer Speed)
2	A (Concentration of Coagulant Added)
3	B (Treatment Temperature)

The gradients of each line depict the effect of the factors that influence the amount of pollutants or the discharge flow rate of the pollutants. The steeper the gradient, the more significant is the factor.

This can be seen from the gradients of the lines plotted where Factor C has the steepest gradient(-14.5) which indicates the most significant effect on the amount of pollutant discharged. It is then followed by Factor A where it has the second steepest gradient(12.5) then lastly Factor B with the least steep gradient(1.5)

Hence, from the graph above, we can see that Factor C(stirrer speed) has the most significant impact on the discharge flow rate of the pollutants as the line is the steepest with the greatest gradient compared to the other 2 factors. This is because as the stirrer speed is of high level(400rpm), the more the wastewater is treated and thus resulting in the lesser amount of pollutant discharged per day and vice versa for the low level of stirrer speed(200rpm)

Factor A(concentration of coagulant added) has the second most significant effect on the discharge flow rate of the pollutants. The lower the concentration of coagulant added(1% by weight), the lesser the discharge flowrate of pollutant and vice versa when the concentration of coagulant is high(2% by weight).

Factor B(treatment temperature) has the least significant effect on the discharge flow rate of pollutants. The lower the temperature(72 degrees Fahrenheit), the lower the discharge flow rate of the pollutant and vice versa for the higher temperature(100 degrees Fahrenheit).

Next, we will determine the interaction effects of 2 different factors. This is to see how the effect of one factor on the response variable varies at different levels of the other factor.

There are a total of 3 interactions that we will be discovering in-depth. Those 3 interactions are between:

• Factors A and B
• Factors A and C
• Factors B and C

Interaction 1: A x B

At LOW B, Average of low A= (5+4)/2=4.5 lb

At LOW B, Average of high A= (30+3)/2=16.5 lb

At LOW B, total effect of A= (16.5-4.5) =12 lb (increase)

 

At HIGH B, Average of low A = (6+5)/2 = 5.5 lb

At HIGH B, Average of high A= (33+4)/2=18.5 lb

At HIGH B, total effect of A = (18.5-5.5) = 13 lb (increase)

From the above graph, we can see that the gradient for both the lines is positive and are different by a small margin. This suggests that there is a minor interaction between Factor A (concentration of coagulant added) and Factor B (treatment temperature). This means that at both low A and high A, Factor B has a similar effect on the water treatment.

Interaction 2:A x C

At LOW C, Average of low A = (5+6)/2 = 5.5 lb

At LOW C, Average of high A = (30+33)/2 = 31.5 lb

At LOW C, total effect of A = (31.5-5.5) = 26 lb

 

At HIGH C, Average of low A = (4+5)/2 = 4.5lb

At HIGH C, Average of high A = (3+4)/2 = 3.5 lb

At HIGH C, total effect of A = (3.5-4.5) = -1lb (decrease)

From the above graph, we can see that the gradients of both lines are different. The line for (-C) is positive while the line for (+C) is negative. Therefore, there is a significant interaction between Factor A(concentration of coagulant added) and Factor C(stirrer speed).

Interaction 3: B x C

At LOW C, Average of low B = (5+30)/2 = 17.5 lb

At LOW C, Average of high B = (6+33)/2= 19.5 lb

At LOW C, total effect of B = (19.5-17.5) = 2 lb(increase)

 

At HIGH C, Average of low B = (4 + 3)/2 = 3.5 lb

At HIGH C, Average of high B = (5+4)/2 = 4.5 lb

At HIGH C, total effect of B = (4.5-3.5) = 1 lb(increase)

From the above graph, we can see that the gradient of both lines is positive and are different by a small margin. This suggests that there is a minor interaction between Factor B(treatment temperature) and Factor C(stirrer speed). This means that at both low and high B, Factor C has a similar effect on the water treatment as the gradient shown is similar.

Conclusion: Factor C has the most effective and significant effect on the water treatment, leading to a lower amount of pollutants discharged per day. It is deduced that Factor C has significant interaction with Factor A compared to Factor B.

Fractional Factorial Data Analysis

For this data analysis, we will need to choose 4 runs from the given 8 runs. The runs that I chose are:

• Run 2
• Run 3
• Run 5
• Run 8

These runs were chosen because all the 3 factors were found in both low and high levels and it occurs the same number of times (2 times). I believe these designs are balanced designs for the experiment and are orthogonal.

The same steps as the full factorial analysis are carried out for this fractional analysis as well, but with the chosen 4 runs only.

Plot a line graph of the discharge flow rate of pollutants vs the factor levels using the average values that we have collected from each factor of high and low levels.

Ranking of Factors from Most Significant to Least Significant: C>A=B

Ranking	Factors
1	C (Stirrer Speed)
2	A (Concentration of Coagulant) B (Treatment Temperature)

The gradients of each line depict the effect of the factors that influence the amount of pollutants or the discharge flow rate of the pollutants. The steeper the gradient, the more significant is the factor.

This can be seen from the gradients of the lines plotted where Factor C has the steepest gradient(-14) which indicates the most significant effect on the amount of pollutant discharged. It is then followed by Factor A (+12) and Factor B (-12) with the same gradient which shows the same level of significance on the amount of pollutant discharged.

Hence, from the graph above, we can see that Factor C(stirrer speed) has the most significant impact on the discharge flow rate of the pollutants as the line is the steepest with the greatest gradient compared to the other 2 factors. This is because as the stirrer speed is of high level(400rpm), the more the wastewater is treated and thus resulting in the lesser amount of pollutant discharged per day and vice versa for the low level of stirrer speed(200rpm)

Factor A(concentration of coagulant added) and Factor B(treatment temperature) has the same level of significant effect on the discharge of pollutant flowrate. The lower the concentration of coagulant added(1% by weight), the lesser the discharge flowrate of pollutant and vice versa when the concentration of coagulant is high(2% by weight). The lower the temperature(72 degrees Fahrenheit), the higher the discharge flow rate of the pollutant and vice versa for the higher temperature(100 degrees Fahrenheit).

Conclusion: The most significant factor still remains to be Factor C in the fractional data analysis. However, there is a slight difference in the significance of Factor A and B in this analysis because they both are of the same significance. This may be due to the fact that the fractional data analysis has lesser data points as compared to the full analysis. Hence, this may affect the accuracy of each factor. However, the trend found is similar in both full and fractional factorial as Factor C has the highest significance.

Link for Full and Factorial Data Analysis:

https://docs.google.com/spreadsheets/d/1q6O_wc_jpmL0EKpOe4hheJqy5seVnEsF/edit?usp=sharing&ouid=103152770832346110694&rtpof=true&sd=true