Design of Engineering Experiments
IEE 572 (Spring 2006)
Term Project
Submitted by:
· Ankit Mishra
· Jonathan Stahlhut
· Sadik Kucuksari
· Sanjay Patil
· Surendra Puchalapalli
· Yan Ma
TABLE OF CONTENTS
1. Introduction 3
2. Objective of the Experiment 3
3. Choice of Factors, Levels and Ranges 3
4. Experimental Setup 4
5. Experiment Factors and Levels 6
6. Selection of Response Variable 6
7. Choice of experimental design 7
8. Performing the experiment 7
9. Statistical Analysis of Data 10
9.1 Half-Normal Plot for Effect Estimate 10
9.2 Analysis of Variances (ANOVA) 11
9.3. Model Adequacy Test 14
9.3.1. Normality Assumption 14
9.3.2 Independence Assumption 15
9.3.3. Constant Variance Assumption 16
9.4. Analysis of Effects in the Model 18
9.4.1 Main Effects 18
9.4.2. Interaction Effects 21
9.5. Box-Cox Transformation 24
10. Conclusions 25
1. Introduction
After natural uranium is taken directly out of the ground, it is milled before it is enriched. The enrichment process allows for PWR and BWR nuclear reactors to generate electricity in the United States. When the natural uranium is milled, the uranium is sometimes in the form of small balls or pellets that are transported to an enrichment facility. At the enrichment facility, the uranium is loaded onto a conveyor belt that transports the uranium to a room to where it is enriched. Since this natural uranium is radioactive, it emits alpha particles in which the range of these particles can be calculated. The enrichment facility is having problems with some of the sensors used in and around the conveyor belt because the small balls of uranium roll off of the conveyor belt and roll close enough to the sensors that the range of an alpha particle is enough to cause the sensors to malfunction. The enrichment facility does not have the capability to add walls to stop the uranium balls from rolling. Their only solution is to redesign the conveyor based on the belt angles, height of the next level of conveyor belt and facility floor surface. Therefore, the enrichment facility wants to run some tests in order to see how far these balls can roll. The facility has provided a list of important parameters that vary along the length of conveyer belt. Since performing experiments with uranium balls is not feasible, an alternative experimental setup is to be built. Material properties of iron balls can be safely assumed to represent uranium balls for the experiment.
2. Objective of the Experiment
The objective of the experiment is to identify the factors affecting the distance traveled by the iron ball under different conditions and to obtain an optimized combination which gives the minimum rolling distance.
3. Choice of Factors, Levels and Ranges
The list of factors for the experiment is based on the factors provided by the facility. The factors governing the rolling distance of the uranium ball are the dimensions of the uranium ball, the angle of inclination of the conveyor side board, the height of the conveyer form the floor and the floor surface on which the balls roll. The uranium balls are a homogeneous material with uniform density. Therefore, different balls of same physical dimensions essentially weigh the same. The options that facilities have to cover their floor surface are plastic matting or carpeting.
Since iron balls are being used in the experiment as a substitute of uranium balls, an important factor governing the rolling distance would be their dimensions. Another factor of importance would be the inclination of the surface on which the iron balls are rolled. The balls can be released from different height to consider differences in the height of conveyer belt from the floor. Homogeneous iron balls of similar dimensions and weight as of uranium are used for conducting experiments. As listed by the facility plastic matting and carpet would be two surfaces for the experiment.
4. Experimental Setup
Figure 1 illustrates the experimental arrangement. The four different types of factors are considered for the experiment is shown in Table 1. Two different levels are considered for each factor.
Figure 1: Experimental Arrangement
Where:
Surface 1 is plastic matSurface 2 is carpet
α1 is 30o
α2 is 60o / Ball 1 is big
Ball 2 is small
H1 is 6 inches
H2 is 3 inches
Figure 2 Experiment of Plastic Matting
Figure 3 Experiment on Carpet
5. Experiment Factors and Levels
Table 1 Experiment Factors and Levels
Factor / Description / Low level / High level / TypeA / Floor / Plastic Mat / Carpet / Categorical
B / Angle of inclination
(degrees) / 30 / 60 / Numeric
C / Size of iron ball / Small / Big / Categorical
D / Height of release (inches) / 3 / 6 / Numeric
Constant factors:
· Type of surface (board) from which the iron ball was released.
· No pull or push force was applied to the iron ball while releasing it.
· Experiment under same environmental conditions like wind, temperature etc...
· The same measure tape was used to measure the distance traveled by ball.
Nuisance Factors:
· Inherent variability in the equipment.
· Friction between iron and floor surface (within process variation).
Blocks:
· As the distance traveled by the iron ball can be affected by the person releasing the ball. Therefore four persons were selected to do individual runs of the experiment. As each person was considered as a block, thus four blocks has been considered. However no person applied any extra push or pull force to the ball, while releasing. The ball was simply released from the inclined surface.
6. Selection of Response Variable
Distance (inches) traveled by the iron ball has been selected as the response variable for the experiment. The distance is measured from the base of the board to the point where iron ball stops.
7. Choice of experimental design
Depending on the above design parameters, four replicate 24 completely randomized block design has been selected to do the experiment. Four different persons were selected as a block in order to reduce the variability that might affect the results. The choice of blocking is also attributed to eliminating the known and controllable factor that is diet in the particular experiment. Thus, we can systematically eliminate its effect on the statistical comparisons among treatments (Design and Analysis of Experiments, D.C. Montgomery, 2000). The experiment is completely randomized to guard against the unknown and uncontrollable factors. Hence, four persons, each in one block and four replicates are chosen for the design
The choice of number of replicates had been decided with the help of design expert. A replicate size of two indicates a 2 standard deviation of 93.7% and replicate size of 3 indicates 99.6 % at 95% confidence interval. Since a higher standard deviation reflects better difference in variability, we therefore have performed a four replicate 24 completely randomized block design.
8. Performing the experiment
The experiment was conducted in the Engineering Research Center at Arizona State University, Tempe. Before running the original experiment, few pilot runs were carried out to observe the response variable and to check the variability in the system. These runs provided consistency in the experimental data. The experiment was conducted in a random order as planned.
The ball was released from the inclined surface during each run order (for example: on carpet at 30 degree and at height of 3 inches; the big ball was released). The distance traveled by the ball in each block and replicate was considered as the response variable. Tables 2.1 and 2.2 show the summary of the experiment data entered in the design experts:
Table 2.1 Experimental Data
Standard Order / Run Order / Block / Floor Type / Angle of Inclination (degree) / Ball Size / Height of release (inches) / Distance (m)1 / 5 / Block 1 / Carpet / 30 / Small / 6 / 52
2 / 16 / Block 1 / Plastic / 30 / Small / 3 / 26.5
3 / 12 / Block 1 / Carpet / 30 / Big / 6 / 70.2
4 / 13 / Block 1 / Plastic / 60 / Small / 6 / 26
5 / 11 / Block 1 / Plastic / 30 / Small / 6 / 53.1
6 / 7 / Block 1 / Carpet / 60 / Small / 3 / 19
7 / 4 / Block 1 / Carpet / 60 / Big / 6 / 33.7
8 / 14 / Block 1 / Carpet / 60 / Small / 6 / 27
9 / 3 / Block 1 / Plastic / 30 / Big / 3 / 35.4
10 / 10 / Block 1 / Carpet / 30 / Small / 3 / 25.4
11 / 1 / Block 1 / Plastic / 60 / Big / 6 / 36.1
12 / 8 / Block 1 / Plastic / 60 / Small / 3 / 18.4
13 / 15 / Block 1 / Carpet / 30 / Big / 3 / 32.5
14 / 2 / Block 1 / Plastic / 60 / Big / a / 21.5
15 / 9 / Block 1 / Plastic / 30 / Big / 6 / 63.3
16 / 6 / Block 1 / Carpet / 60 / Big / 3 / 23
17 / 32 / Block 2 / Carpet / 60 / Big / 6 / 38.8
18 / 26 / Block 2 / Carpet / 30 / Small / 3 / 24.5
19 / 20 / Block 2 / Plastic / 30 / Big / 3 / 34.2
20 / 19 / Block 2 / Plastic / 60 / Small / 3 / 20.6
21 / 25 / Block 2 / Plastic / 30 / Big / 6 / 67.2
22 / 27 / Block 2 / Plastic / 60 / Big / 6 / 39.2
23 / 28 / Block 2 / Carpet / 30 / Big / 3 / 36.6
24 / 30 / Block 2 / Carpet / 30 / Big / 6 / 64.4
25 / 18 / Block 2 / Carpet / 30 / Small / 6 / 45.4
26 / 24 / Block 2 / Plastic / 30 / Small / 3 / 22.1
27 / 23 / Block 2 / Plastic / 60 / Big / 3 / 27
28 / 17 / Block 2 / Plastic / 30 / Small / 6 / 52
29 / 29 / Block 2 / Plastic / 60 / Small / 6 / 21.8
30 / 22 / Block 2 / Carpet / 60 / Small / 3 / 17.5
31 / 31 / Block 2 / Carpet / 60 / Big / 3 / 22.6
32 / 21 / Block 2 / Carpet / 60 / Small / 6 / 27.6
33 / 47 / Block 3 / Carpet / 60 / Big / 3 / 22.5
34 / 43 / Block 3 / Carpet / 60 / Small / 3 / 16.4
35 / 42 / Block 3 / Plastic / 30 / Small / 6 / 49.5
36 / 46 / Block 3 / Plastic / 30 / Big / 6 / 64.9
37 / 45 / Block 3 / Plastic / 60 / Big / 3 / 24
38 / 37 / Block 3 / Plastic / 30 / Small / 3 / 22.5
39 / 41 / Block 3 / Carpet / 30 / Small / 6 / 48.4
40 / 48 / Block 3 / Carpet / 30 / Big / 6 / 65.5
41 / 40 / Block 3 / Plastic / 30 / Big / 3 / 36.7
42 / 33 / Block 3 / Plastic / 60 / Small / 3 / 24
Table 2.2 Experimental Data Continued
Standard Order / Run Order / Block / Floor Type / Angle of Inclination (degree) / Ball Size / Height of release (inches) / Distance (m)43 / 39 / Block 3 / Plastic / 60 / Small / 6 / 23.6
44 / 38 / Block 3 / Plastic / 60 / Big / 6 / 37.5
45 / 36 / Block 3 / Carpet / 30 / Big / 3 / 35
46 / 34 / Block 3 / Carpet / 30 / Small / 3 / 20.5
47 / 44 / Block 3 / Carpet / 60 / Small / 6 / 25.7
48 / 35 / Block 3 / Carpet / 60 / Big / 6 / 35.9
49 / 63 / Block 4 / Carpet / 30 / Big / 3 / 31.7
50 / 49 / Block 4 / Plastic / 60 / Small / 6 / 25.1
51 / 52 / Block 4 / Plastic / 30 / Big / 6 / 62.4
52 / 56 / Block 4 / Plastic / 30 / Small / 3 / 25.1
53 / 62 / Block 4 / Carpet / 30 / Small / 6 / 49
54 / 57 / Block 4 / Carpet / 60 / Big / 3 / 22.3
55 / 64 / Block 4 / Carpet / 30 / Big / 6 / 67.3
56 / 53 / Block 4 / Carpet / 30 / Small / 3 / 22.2
57 / 59 / Block 4 / Plastic / 30 / Big / 3 / 34.3
58 / 54 / Block 4 / Plastic / 60 / Big / 3 / 20.3
59 / 50 / Block 4 / Plastic / 60 / Big / 6 / 36
60 / 61 / Block 4 / Plastic / 30 / Small / 6 / 46.8
61 / 55 / Block 4 / Plastic / 60 / Small / 3 / 19.7
62 / 51 / Block 4 / Carpet / 60 / Small / 6 / 24.3
63 / 60 / Block 4 / Carpet / 60 / Big / 6 / 39.5
64 / 58 / Block 4 / Carpet / 60 / Small / 3 / 23
9. Statistical Analysis of Data
The statistical analysis software package (Design Expert) has been used to analyze the data collected from the experiment. The data was analyzed through certain graphs and model adequacy testing and confidence interval estimation procedures were carried out. The analysis includes ANOVA, residual analysis and model adequacy checking, and regression analysis.
9.1 Half-Normal Plot for Effect Estimate
Figure 4 below shows the half-normal plot, which shows the effects of factors considered for the analysis. Based on this graph, where the response variable is distance traveled by iron ball, the factors that lie along the line are shown negligible.
Figure 4 Half-Normal Plot for effect estimate
Inference: From the figure it can be concluded that three main effects B, C and D and a two-factor interaction BC, BD, and CD are significant.
9.2 Analysis of Variances (ANOVA)
The ANOVA output is shown below. Based on the response variable, distance, it shows that the main effect B, C, D and BC, BD, and CD interaction is significant.