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Project ReportDesign of Experiments – IEE 572

Experimental Design and Analysis

Exploring Efficient Heating Containers

For Use in Microwave Ovens

Instructor: Dr. Douglas Montgomery

Presented byCan CuiHe Peng

Zohair Zaidi (All from 4:30class)

1. Executive Summary

The objective of our experiment is to seek the combination of container factors that minimizes time consumed to heat the substance inside to reach desired temperature.

Our choice of factors: material type, container shape, material color, and cover status, composed the treatments investigated in our experiment. The nature of the experiment resulted in thermometer-measured observations of temperature as response variables which are obtained by heating the substance in the container in the same time interval.

The experimental design chosen was a factorial 24 Randomized Complete Block Design (RCBD) with one block factor. The design matrix for the experiment was generated through a Custom Design in JMP and Design-Expert and the Fit Model analysis tool provided the output to support the documented conclusions.

2. Problem Recognition

2.1 IntroductionFor many decades now since the early 1950’s, microwave ovens have been commonly used in household kitchens to heat up and cook a variety of food and liquids. The microwave oven started off as being a giant 6 ft 750 lb machine but after many years of research and design improvements, it now finds itself being able to be placed in almost any kitchen décor. However, despite all the research that has gone into improving the design of the microwave oven, little thought has been given to the selection of the most efficient choice of material used as a container for heating. In this experiment we will be attempting to optimize the conditions for the ideal heating container that can be used in a microwave and provide the best statistical results. 2.2 MotivationSince currently most people do not follow any good design for heating substances in the microwave, a lot of energy and time is wasted which has a huge negative impact on the environment. Through the implementation of our designed experiment, we can potentially save a lot of energy and revolutionize the way people heat up their food and drinks.2.3 ObjectiveThe objective of our experiment is to determine which factors significantly affect the efficiency of heating inside a microwave. Once those factors are recognized, then we want to optimize the factors such that the amount of time used to heat a substance is minimized and thereby the amount of energy consumed in heating is reduced. 3. Choice of Factors

The four container factors we hypothesize are most important with respect to overall optimization include: material type, container shape, material color, and cover status. These four factors compose the experimental factors we wish to study in our experiment. Table 1 list these factors along with the chosen levels for each. The potential factors that we are considering for our design are as follows:

Table 1: Experimental factors and levelsFactor Level 1 Level 2

Type of Material Plastic GlassShape Cylinder RectangleColor Clear OpaqueCover Open Closed

4. Description of Factors

4.1 Type of MaterialThe two types of materials we chose to test in this experiment are plastic and glass. These materials were chosen because they are most commonly used as containers among housewives. We think the type of material makes a significant difference in the efficiency of heating because of the difference in material absorption of microwaves.

4.2 ShapeThe shapes of container we chose to test are cylinder and rectangle. These shapes were chosen due to them being used in drinking containers and in sandwich boxes. We believe the shape will have a significant difference in heating efficiency because since they are very distinct in form and open area, the directions by which the microwaves will reach the substance will differ significantly, potentially affecting the rate of heating.

4.3 ColorWe will be using two types of colored containers – clear and opaque. The color of the container should make a significant difference in heating efficiency because of the relationship between the color of a material and the wavelengths it absorbs or transmits. Therefore it is possible that a certain type of color tends to absorb microwaves better than others.

4.4 CoverThe last factor chosen was whether or not the container will be covered. It is known that when a container is covered, it will keep heat from escaping from the container. But at the same time, it may also impede the flow of additional microwaves into the container. Since the relative rates of both are not known, it is imperative to test this factor and determine which one plays a more important role in heating efficiency.

5. Selection of Response Variable

Temperature is the response variable selected that will be used to determine which combination of factors allow for most efficient heating inside of a microwave. The initial temperature will be held constant for all runs and then immediately after the samples are processed in the microwave, the final temperature will be measured for all samples using a digital thermometer to ensure accuracy of the measurements.

The experimental design chosen was a factorial 24 Randomized Complete Block Design (RCBD) with one block factor, the type of substance being heated. The substances chosen in the block were water and milk to ensure that the type of substance does not interact with the container and the change in temperature are truly due to the effect of the factors relating to the container. The design matrix for the experiment was generated through a Custom Design in JMP. The experimental worksheet is provided below:

Table 2: JMP Experimental Design Worksheet

Run Block Material Shape Color CoverTemperatur

e1 Water Glass Cylinder Opaque Closed .2 Water Glass Rectangle Clear Closed .3 Water Plastic Rectangle Clear Open .4 Water Plastic Cylinder Opaque Open .5 Water Plastic Rectangle Opaque Closed .6 Water Plastic Cylinder Opaque Closed .7 Water Plastic Rectangle Clear Closed .

8 Water Plastic Cylinder Clear Closed .9 Water Glass Cylinder Clear Closed .10 Water Glass Cylinder Clear Open .11 Water Plastic Rectangle Opaque Open .12 Water Plastic Cylinder Clear Open .13 Water Glass Rectangle Opaque Open .14 Water Glass Cylinder Opaque Open .15 Water Glass Rectangle Clear Open .16 Water Glass Rectangle Opaque Closed .17 Milk Glass Cylinder Clear Open .18 Milk Glass Rectangle Opaque Open .19 Milk Plastic Cylinder Opaque Closed .20 Milk Glass Rectangle Clear Closed .21 Milk Glass Rectangle Clear Open .22 Milk Plastic Cylinder Clear Open .23 Milk Plastic Cylinder Opaque Open .24 Milk Glass Cylinder Opaque Open .25 Milk Plastic Rectangle Clear Open .26 Milk Glass Rectangle Opaque Closed .27 Milk Glass Cylinder Opaque Closed .28 Milk Plastic Cylinder Clear Closed .29 Milk Plastic Rectangle Opaque Open .30 Milk Glass Cylinder Clear Closed .31 Milk Plastic Rectangle Opaque Closed .32 Milk Plastic Rectangle Clear Closed .

6. Performing the experiment

We prepared eight categories of containers which are listed in Table 2. The measuring tool we used is an electronic infrared thermometer which has a decimal accuracy. Please see the Appendix for the pictures of containers and microwave oven and electronic infrared thermometer. There are two blocks, one is water, and the other is milk. We performed the experiment in the same day. First, we did experiment with water which is contained in a big plastic container to ensure unique water resource and to minimize the variation of the water temperature. Moreover, we measured the temperature of water before each run so we can keep records of how much the temperature differs after being heated instead of a single temperature value after heating which makes the experiment more precise. Therefore, response in this experiment is the temperature difference instead of the temperature after heating. To minimize the variation of microwave oven temperature between each run, we cooled down the environment inside the oven by fanning after heating, and started next run until the temperature dropped to normal value. For the experiment with milk, we did exactly the same routine as with water.The result of experiment is shown in table 3.

Table 3: Measured Data Result

Run Block Material Shape Color CoverBefor

e Afte

r Difference1 Water Glass Cylinder Opaque Closed 26.4 85.2 58.82 Water Glass Rectangle Clear Closed 24.4 75.7 51.33 Water Plastic Rectangle Clear Open 24.1 83 58.94 Water Plastic Cylinder Opaque Open 23.6 85.2 61.65 Water Plastic Rectangle Opaque Closed 26.7 81 54.3

6 Water Plastic Cylinder Opaque Closed 23.5 85.5 62.07 Water Plastic Rectangle Clear Closed 22.4 83.6 61.28 Water Plastic Cylinder Clear Closed 23.0 82.9 59.99 Water Glass Cylinder Clear Closed 23.8 80.1 56.310 Water Glass Cylinder Clear Open 23.6 83.3 59.711 Water Plastic Rectangle Opaque Open 26.7 76 49.312 Water Plastic Cylinder Clear Open 23.5 84.9 61.413 Water Glass Rectangle Opaque Open 23.3 81.9 58.614 Water Glass Cylinder Opaque Open 26.5 85.9 59.415 Water Glass Rectangle Clear Open 25.0 77.5 52.516 Water Glass Rectangle Opaque Closed 23.9 78.1 54.217 Milk Glass Cylinder Clear Open 24.0 86.6 62.618 Milk Glass Rectangle Opaque Open 21.0 87.5 66.519 Milk Plastic Cylinder Opaque Closed 13.4 82.3 68.920 Milk Glass Rectangle Clear Closed 14.9 74 59.121 Milk Glass Rectangle Clear Open 16.6 79 62.422 Milk Plastic Cylinder Clear Open 16.6 98.4 81.823 Milk Plastic Cylinder Opaque Open 16.2 85.4 69.224 Milk Glass Cylinder Opaque Open 19.1 81.4 62.325 Milk Plastic Rectangle Clear Open 19.2 86.2 67.026 Milk Glass Rectangle Opaque Closed 18.7 84.3 65.627 Milk Glass Cylinder Opaque Closed 19.3 83.3 64.028 Milk Plastic Cylinder Clear Closed 19.5 95.5 76.029 Milk Plastic Rectangle Opaque Open 15.2 76.9 61.730 Milk Glass Cylinder Clear Closed 19.9 80.5 60.631 Milk Plastic Rectangle Opaque Closed 17.2 80.1 62.932 Milk Plastic Rectangle Clear Closed 18.5 82.7 64.2

7. Statistical analysis of the data (Using both JMP and Design-Expert)

7.1 Temperature Response

Table 4: Summary of FitRSquare 0.846565

RSquare Adj 0.773501

Root Mean Square Error 3.194174Mean of Response 61.69375

Observations (or Sum Wgts) 32

The R-squared value indicates how much of the total variation is explained by the regression model. It is possible sometimes this value to be inflated due to large number of factors included in the model, therefore a R-squared adjusted value is also calculated which takes into consideration the number of factors included in the model. In this case, 77.3% of the variation can be explained by the constructed model of the chosen factors, which is relatively good and gives confidence to the models ability in capturing the source of variation in the factors that we have chosen.

7.2 Factor Evaluation

Table 5: Parameter EstimatesTerm Estimate Std Error DFDen t Ratio Prob>|t|Intercept 61.69375 4.23125 1 14.58 0.0436*Materials[Plastic] 2.075 0.564656 20 3.67 0.0015*Shape[Cylinder] 2.3375 0.564656 20 4.14 0.0005*Colors[Clear] 0.4875 0.564656 20 0.86 0.3982Cover[Open] 0.4875 0.564656 20 0.86 0.3982Materials[Plastic]*Shape[Cylinder] 1.49375 0.564656 20 2.65 0.0155*Materials[Plastic]*Colors[Clear] 2.04375 0.564656 20 3.62 0.0017*Materials[Plastic]*Cover[Open] -0.39375 0.564656 20 -0.70 0.4936Shape[Cylinder]*Colors[Clear] 0.26875 0.564656 20 0.48 0.6393Shape[Cylinder]*Cover[Open] 0.23125 0.564656 20 0.41 0.6865Colors[Clear]*Cover[Open] 0.61875 0.564656 20 1.10 0.2862

From the p-values generated by JMP using the custom factorial model, it is evident that the significant factors in this experiment are Materials, Shape, Materials & Shape interaction, and Materials & Color interaction. Once the p-value goes below a certain value determined from the t-table and degrees of freedom, the factors can be declared as being significant.

ANOVA for selected factorial model

Table 6: Analysis of variance table [Partial sum of squares - Type III]Source Sum of

Squares dfMean

SquareF

Valuep-value

Prob > FBlock 575.17 1 575.17Model 527.03 5 105.41 11.36 < 0.0001significantA-Materials 138.33 1 138.33 14.91 0.0007B-Shape 174.53 1 174.53 18.82 0.0002C-Colors 7.74 1 7.74 0.83 0.3698AB 71.40 1 71.40 7.70 0.0103AC 135.03 1 135.03 14.56 0.00 08Residual 231.87 25 9.27Cor Total 1334.07 31

The results are further corrobarated by running the analysis in Design Expert and from the F-values, the same factors can be seen as significant. The F-statistic is calculated by taking the proportion of the mean square of the factor by the mean square of the residual. Thus, the higher the value of mean square, the more significant the factor can be seen to be. In this case, it appears that the main effect Shape has the highest mean square and perhaps is the most significant factor.

Design-Expert?SoftwareTemperature

Error estimates

Shapiro-Wilk testW-value = 0.911p-value = 0.286A: MaterialsB: ShapeC: ColorsD: Cover

Positive Effects Negative Effects

Half-Normal Plot

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Figure 1 Half-Normal Plot

The half-normal probability plot is a visual way of identifying which factors are singificant in an experiment. The plot takes the absolute value of effect estimates and plots it against their normal probabilites in a cumulative manner. Those effects which are significant will be plotted far away from the straight line that crosses the origin. Thus the same factors which were identified through the p-values and F-statistic are found to be significant here, A (Materials), B(Shape), AB(Material/Shape Interaction), and AC(Materials/Colors interaction.

Final Equation in Terms of Coded Factors:

Temperature =+61.69+2.08 * A+2.34 * B-0.49 * C+1.49 * A * B-2.05 * A * C

Once the significant factors are identified, an equation can be constructed for the model using the effect coefficients calculated in the table above. This equation can then be used to predict values according to generate prediction values that will later be used in residual analysis to test the validity of the assumptions made about this experiment.

7.3 Effect of blocking

Table 7: REML Variance Component EstimatesRandom Effect Var Ratio Var Component Std Error 95% Lower 95% Upper Pct of TotalRandom Block 3.4470394 35.169281 50.63908 -64.08149 134.42005 77.513Residual 10.20275 3.2263928 5.9718276 21.276169 22.487Total 45.372031 100.000

-2 LogLikelihood = 150.52238998

According to the whole effect of the system, the block plays a really important role during the experiment.If let the system ignore the block affect and just put 4 factors in the system, the R-squared adjusted value is lower than 30% which reduce the confidence of this model a lot and the majority of variance would not be explained. So setting up this block is a right choice.

Effect Details Random Block

Table 8: Least Squares Means TableLevel Least Sq Mean Std ErrorMilk 65.849647 0.79498041Water 57.537853 0.79498041

Figure 2 LS Means and Block Plot

This figure indicates that the material of liquids makes a lot of difference in the change of temperature. This is reasonable in physics. The heat capacities of milk and water are 0.94 vs 1.0. According to the equation of heat

capacity, which shows the linear relative between capacity and temperture. Since milk have lower heat capacity, and assume they get same quantity of heat, milk should have higher change of temperature than water should have. The result of experiment is fit for the theory.

7.4 Residual Analysis Design-Expert?SoftwareTemperature

Color points by value ofTemperature:

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Figure 3 Normal Plot of Residuals

Analyzing the residuals is an important way to verify the assumptions to determine whether the results from the factorial analysis can be depended upon. The first assumption that is made is that the data comes from a population of normal distribution. To check this assumption, a normal probability plot is constructed from the residuals. If the assumption is met, then the residuals should all fall on a straight line with little deviation. In this case, the majority of the residuals seems to fall in line with only few points which seem to be outliers and should not significantly affect our data, therefore we can conclude there are no alarming problems with normality.



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Figure 4 Residuals vs. Predicted Plot



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Figure 5 Residuals vs. Run Plot

A second way to confirm that the model chosen is correct and that the assumptions is met is to plot the residuals by the predicted values and by the run. The predicted values are generated from the equation that is created

from the effect coefficients of the significant factors as was shown previously. If the assumptions are met, then both plots should be structureless and show no apparent pattern or correlation between the points. Indeed such is the case for both plots, with only slight deviation in the first plot, further confirming that our model is correct and that there is nothing major to worry about in terms of reliability of our analysis.



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Figure 6 Predicted vs. Actual Plot

Another way to see how well the model fits the data is to directly compared the predicted values versus the actual values that were obtained by the experiment. If the model is a first-order model and there is a complete fit, then the points should all fall along the straight line. In this case, there is some variance between the predicted and actual values, but relatively speaking there is a good fit. Usually there is always some skewing that occurs at the extreme ends, in this case at the extremely low and high temperatures, hence the appearance of an s-shaped curve.

7.5 Variance Analysis


Color points byStandard Order

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Figure 7 Residuals vs. Block Plot



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Figure 8 Residual vs. Materials Plot



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Figure 9 Residual vs. Shape Plot



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Figure 10 Residual vs. Colors Plot



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Figure 11 Residual vs. Cover Plot

The Figure above shows nothing very unusual in the experiment. All the residual in absolute value is lower than the standardized value 3.00. And the distribution between positive value and negative value is random and has no tendency to be one side. Have to mention that there is a mild tendency to increase or decrease from left side to right side, however, the problem is not severe enough to have a dramatic impact on the analysis and conclusions.

7.6 Factors Interaction

Design-Expert?SoftwareFactor Coding: ActualTemperature

Design Points

X1 = A: MaterialsX2 = B: Shape

Actual FactorsC: Colors = OpaqueD: Cover = Closed

B1 RectangleB2 Cylinder

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Glass Plastic

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Figure 12 Material-Shape Interaction Plot

Interaction plots generated by JMP are a good way to see the effect of each factor relative to another factor. Since the Material and Shape factors were significant they are varied and the other two factors that are not

significant are held constant. It can be seen that in the case of cylinder made of plastic, there is an increase in temperature whereas in the case of glass, it decreases for plastic.


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Actual FactorsC: Colors = ClearD: Cover = Closed


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If the color is then changed from opaque to clear, we see further improvements in the slope between the glass and plastic, indicating that results are better when using a container that is both plastic and clear in color. It’s

also interesting to note that the slope changes to positive in the case of the rectangle, though it is still quite below the results obtained using a cylinder.


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Actual FactorsC: Colors = OpaqueD: Cover = Open


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When the container is then open and opaque, there seems little difference between the glass and plastic cases, indicating this combination is not effective in improving temperature results.


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In the case of clear and open, we see that in the glass case there is possible interaction that may occur, but it is not close enough to be declared an interaction.


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Figure 16 Material-Colors Interaction Plot

When we then switch the constant variable to become the shape and vary the material and color, we can see that there is a clear interaction that is occurring between the material and color. This is a clear example of why it’s important to look at interaction plots to see which factors are affected by other factors.

7.7 The Best Combination of Our Experiment

As Figure shown below, the highest difference of temperature 70.1458 is obtained by the combination of plastic as material, clear as color and cylinder as shape and open as cover. As we have concluded, among all four factors, only material, shape, interaction of material and shape and interaction of material and color are significant. Therefore, whether the cover is open or closed does not make a lot of influence to the temperature increasing. This can also be noticed by directly observing the experimental result:

Table 8: Partial Measured Data Result

Run Block Material Shape Color CoverBefor

e Afte

r Difference7 Water Plastic Cylinder Clear Closed 23.0 82.9 59.912 Water Plastic Cylinder Clear Open 23.5 84.9 61.422 Milk Plastic Cylinder Clear Open 16.6 98.4 81.828 Milk Plastic Cylinder Clear Closed 19.5 95.5 76.0

There is only 1.5°F and 5.8°F difference respectively in experiments with water and milk and as the number of replicates increases, the difference is supposed to decrease. In conclusion, the best combination of factor levels is “plastic, cylinder, clear and open”.

Design-Expert?SoftwareFactor Coding: ActualTemperatureX1 = A: MaterialsX2 = B: ShapeX3 = C: Colors

Actual FactorD: Cover = Open

CubeTemperature

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Figure 17 Cube

8. Conclusion

In this experiment we attempted to optimize the conditions for the ideal heating container that can be used in a microwave oven and provide the best statistical results. The experimental design chosen was a factorial 24 Randomized Complete Block Design (RCBD) with the type of substance being heated as a block factor. We analyzed the outputs in JMP and Design Expert which provided us several comprehensive analyses such as, ANOVA table, half-normal probability plot, normal plot of residuals, residuals vs predicted and residuals vs runs… Finally, we concluded that the best combination of factor levels is “plastic, cylinder, clear and open”.

This experiment is very meaningful because through the implementation of our designed experiment, we find the most efficient way that can potentially save a lot of energy and revolutionize the way people heat up their food and drinks. Due to the limitation of our time and energy, we did only one replicate. In the future, we could do more replicates and also add more factors thus make our experimental results more reliable.

9. Appendix:

Microwave oven

Thermometer

Some of Containers

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