ChemEng 4C03

Design and Analysis of Experiments Project

Instructor: Kevin Dunn T.A.: Ian Washington

Submitted by:

Alexander Cushing – 0771472 Abdul Shehata – 0647787

Greg Voloshenko – 0659197

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Table of Contents 1. Objective ..................................................................................................................................... 3

2. Factors Influencing Outcome ...................................................................................................... 3

3. Disturbances ................................................................................................................................ 4

4. Experimental Program ................................................................................................................ 5

5. Executing Experimental Program ............................................................................................... 6

6. Analyze Experimental Results .................................................................................................... 7

6.1 Data Analysis ........................................................................................................................ 7

6.2 Discussion ........................................................................................................................... 11

7. Conclusions ............................................................................................................................... 12

8. Bibliography ............................................................................................................................. 14

9. Appendix A: R Code ................................................................................................................. 15

10. Appendix B: Raw Data ........................................................................................................... 17

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1. Objective

The objective of this experiment was to see how different factors affect the time necessary to

melt an ice cube of predetermined mass in a predetermined amount of water. In particular, 6

factors were varied: quantity of salt in water, quantity of sugar in water, temperature, pot type,

operator and type of water. All of these factors were compared to determine which one was the

most effective at melting ice and to see their interactions.

2. Factors Influencing Outcome

Each of the six factors have 2 levels – low and high. Below, Table 1 presents the 6 factors along

with their respective low, -1, and high levels, 1.

Table 1. Factors and their corresponding levels

All impurities were added to 3 cups of water in a pot. The quantities of salt and sugar were

measured with a table spoon. The same table spoon was used for measuring the quantities of

both the salt and the sugar; this ensured that the same quantities were always measured. Also,

the table spoon was washed and dried after each measurement to prevent cross contamination

between salt and sugar. For both the salt and the sugar, the low value was 0 mL and the high

value was 15 mL. In the experiment, it was predicted that adding either salt or sugar to the water

would disrupt the dynamic freezing-melting equilibrium around the ice cube. Water melting

from the cube would diffuse into the salt or sugar water instead of refreezing onto the cube

(Topper 2011). This would result in increased melting of the cube, and the water temperature

would continue to fall until the freezing-melting equilibrium is re-established (Topper 2011). In

-1 0 mL

1 15 mL

-1 0 mL

1 15 mL

-1 40 °C

1 60 °C

-1

1

-1

1

-1

1

Factor 1:Salt

Factor 2: Sugar

Factor 3:Temperature

B

Tap

Distilled

Factor 4: Pot

Factor 5: Operator

Factor 6: Water

Greg

Alexander

A

4

other words, it was expected that impurities would lead to a faster melting time since impurities

in water commonly lower the freezing point of water.

The temperature of the water was measured with a cooking thermometer. The thermometer was

placed in the pot of water during heating. At this time, the contaminants had already been added

to the 3 cups of water. Once the water reached the desired temperature the thermometer was

removed from the pot and the ice was added to the water. Additionally, to minimize the effects

between thermometers, only 1 thermometer was used. For the experiment, it was expected that a

higher temperature would lead to a faster melting time since molecules tend to move faster at

higher temperatures.

The experiments occurred in two pots – pot A and pot B. Pot A was a Teflon pot and pot B was

a plain stainless steel pot. Before the pot was placed on the stove, 3 cups of water (750 mL)

along with the prescribed impurities were added to the water. For the experiment, it was

expected that the stainless steel pot would lead to a faster melting time since stainless steel pots

have a more evenly distributed heating pattern than Teflon pots.

The experiments were carried out by two different operators. Each operator measured the water

and the impurities for their experiment and added them to the pot. Further, for each operator’s

experiment, they measured the temperature of the water, added the ice cube to the water and

were the judge of when the ice cube had melted. For the experiment, it was expected that the

operator would not affect the melting time since both operators were expected to follow the same

procedure.

Two types of water were used: distilled water and tap water. President’s Choice distilled water

was used as the distilled water and Hamilton city water was used as the tap water. For the

experiment, it was expected that the tap water lead to a faster melting time than the distilled

water because tap water is expected to have more impurities than distilled water which

commonly result in lowering the freezing point of water.

In terms of interactions, it was expected that all of the factors would interact with each other to

some extent. Specifically, it was expected that there would be strong 2 factors interactions

between: impurities and temperature, impurities and type of water and temperature and type of

water. It is not expected that interactions with 3 or more factors will be strong.

3. Disturbances

Factors that are known to affect the melting time of ice but are not being investigated here are:

Pressure – different pressures are known to affect the evaporation rate of water and thus,

may affect the melting rate of ice. During the experiment, it was assumed that the room

was at a constant pressure. Also, the experiments were carried out when the weather was

the same outside resulting in similar pressures.

Impurity levels in tap water – different impurity levels may be present in the tap water at

different times during the day due to disturbances in the water entering the water

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treatment facility. Cold tap water was used in the experiments to keep the number of

impurities in the water constant and to minimize the number of impurities in the water.

Water temperature – different areas in the pot of water may lead to different temperature

readings due to variations in the heating from the element. As such, the thermometer was

always placed in the same spot in the pot for all the experiments.

4. Experimental Program

For the experiment, a full factorial set of experiments would yield 26 = 64 experiments. The

smallest number of runs that could have been performed to screen for the main effects is 23 = 8.

This is a small number of experiments given both the amount of time for one run and the cost for

one run (10 min/run and about 50 cents/run respectively). A ½ fraction factorial experiment was

performed. Performing a ½ fraction factorial experiment allowed for a higher design resolution

to be reached than if a lower fractional factorial experiment were performed, which would have

caused more confounding. Performing a full factorial set of experiments would have required

too much time.

The experiments were run randomly in parallel on two stove top heating elements. This was

done so that the experiments could be run efficiently in a shorter period of time. As a result, the

list of experiments was blocked into 2 blocks. For each block, the experiments were randomly

ordered. For this a random number generator was used; the random number generator was

primed (ran a few times) before using it so that the numbers would be as random as possible.

The ranges were determined as follows:

Salt: the low level was 0 mL and the high level was 15 mL. Fifteen millitres of salt was

selected as the high level as it created a low concentration solution on par with seawater,

a subject on which there is an abundance of information. The high level of salt results in

an ion concentration of 0.045g/mL.

Sugar: the low level was 0 mL and the high level was 15 mL. Sugar (sucrose) was

selected as an alternate solute to compare against the melting effects of salt. The high

level of sugar results in a concentration of 0.033g/mL.

Temperature: the low level was 40°C and the high level was 60°C. A temperature range

above room temperature was selected to better study the interactions of temperature and

solute miscibility on melting rate.

Pot: the low level was a Teflon coated pot and the high level was a stainless steel pot.

Two different pots were used to determine if pot coating had an effect on melting time.

Operator: the low level was Greg and the high level was Alexander. Two operators were

used to determine if there was a bias in the observations between operators.

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Water: the low level was tap water and the high level was distilled water. Both distilled

water and tap water were used to further compare effects of dissolved impurities on

melting time.

5. Executing Experimental Program

Tables 2 and 3 below present the recorded melting times for each experiment with the

experimental conditions.

Table 2. Melting times for heating element 1 with experimental conditions

A B C D E F y

Salt Sugar Temperature Pot Operator Water Melting Time (s)

3 -1 -1 -1 -1 -1 -1 164.9

5 -1 1 -1 -1 -1 1 197.1

9 1 1 1 -1 -1 1 152.4

11 1 1 -1 -1 1 1 285.7

13 -1 1 1 1 1 -1 83.3

14 -1 -1 -1 -1 1 1 155.1

17 1 1 -1 1 1 -1 224.1

19 1 -1 1 1 1 -1 122.1

20 1 1 1 -1 1 -1 128.1

21 1 -1 1 1 -1 1 135.9

22 1 -1 1 -1 1 1 137.1

24 -1 -1 -1 1 -1 1 207.3

25 1 1 -1 1 -1 1 279.4

26 -1 -1 -1 1 1 -1 107.1

28 -1 1 1 -1 -1 -1 85

29 -1 -1 1 1 -1 -1 98.9

Heating Element 1

Experiment

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Table 3. Melting times for heating element 2 with experimental conditions

Below, Figure 1 presents a picture with all the equipment and materials that were used in the

experiment.

Figure 1. Picture presenting all the equipment used in the experiment

6. Analyze Experimental Results

6.1 Data Analysis

A least squares regression on the data for up to two interaction terms produces the following

model:

A B C D E F y

Salt Sugar Temperature Pot Operator Water Melting Time (s)

1 1 -1 -1 1 -1 -1 305

2 1 -1 -1 -1 -1 1 239.4

4 -1 -1 1 -1 1 -1 87.1

6 1 1 1 1 1 1 118.6

7 -1 1 -1 -1 1 -1 280.8

8 -1 1 1 1 -1 1 105.1

10 -1 1 1 -1 1 1 90.6

12 1 -1 -1 -1 1 -1 262.1

15 -1 1 -1 1 -1 -1 232.9

16 1 1 1 1 -1 -1 182.1

18 -1 1 -1 1 1 1 210.1

23 -1 -1 1 -1 -1 1 130.1

27 1 -1 -1 1 1 1 219.2

30 1 -1 1 -1 -1 -1 85

31 1 1 -1 -1 -1 -1 228

32 -1 -1 1 1 1 1 94.3

Experiment

Heating Element 2

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Interactions with 3 or more factors were not determined because interactions with 3 or more

factors are commonly very small and thus, negligible.

Figure 2 below presents a Pareto plot of the coefficients for the factors from the linear model,

where:

A = Salt; B = Sugar; C = Temperature; D = Pot; E = Operator; and, F = Water.

Figure 2. Pareto plot of model coefficients

Based on the Pareto plot, the most significant factors are: C, A, DE and B. These correspond to

temperature, salt, pot & operator interaction and sugar respectively.

Temperature is clearly the most significant factor with a coefficient of -55.1. This represents a

decrease in melting time for every 1 degree Celsius increase in temperature while holding all

other factors constant.

The next significant factor is the quantity of salt added to the water with a coefficient of 24.2.

This represents an increase in melting time for every 1 mL increase in the quantity of salt added

to the water while holding all other factors constant.

Magnitude of effect

Effe

ct

AECE

D

AFBDEF

DFCD

F

BFCFAD

BEABAC

EBC

B

DEAC

0 10 20 30 40 50

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The coefficient for pot and operator interaction, -16.1, represents a decrease in melting time for

every 1 unit increase in the interaction between the pot and the operator while holding all other

factors constant.

The coefficient for sugar, 10.4, represents an increase in melting time for every 1 mL increase in

the quantity of sugar while holding all other factors constant.

Next, confidence intervals were calculated for each of the factors. They are as follows:

The confidence interval for salt does not span 0 and is not symmetric. Thus, the effect of salt is

significant.

The confidence interval for sugar spans 0 and is not symmetric. Thus, the effect of salt is

significant

The confidence interval for temperature does not span 0 and is very far from 0. Thus, the

confidence interval is not symmetric. Hence, the effect of temperature is very significant.

The confidence interval for the pot spans 0 and is symmetric. Thus, the effect of the pot is

insignificant.

The confidence interval for the operator spans 0 and is not symmetric. Thus, the effect of the

operator is significant but it is not as significant as the effects of temperature, salt and sugar.

The confidence interval for water spans 0 and is roughly symmetric. Thus, the effect of water is

insignificant.

The residuals for the linear model are plotted in the two figures below. Figure 3 is a qq-plot of

the residuals and Figure 4 is a time series plot of the residuals.

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Figure 3. QQ-plot of model residuals

In Figure 3, most of the residuals follow the 45 degree line and are within the boundaries.

Further, there is no strong evidence of non-normality; however, there is a trend in the tails on

both sides.

Figure 4. Residual plot

The residuals in time order show no consistent structure and are randomly scattered around 0,

thus, implying that the data is random.

To test for autocorrelation, Figure 5, an autocorrelation plot, was generated and is presented

below.

-2 -1 0 1 2

-50

05

0

norm quantiles

resid

(mo

de

l)

0 5 10 15 20 25 30

-50

05

0

Index

resid

(mo

de

l)

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Figure 5. Autocorrelation plot of residuals

From looking at Figure 5, there is little correlation between the residuals and they appear

independent of each other.

6.2 Discussion

Initially, it was assumed that adding impurities to the water would result in reduced melting

times. Impurities within the water were expected to increase the melting rate due to diffusion of

cold ice water into solution, as well as from interference to refreezing caused by the presence of

the solute. This was not the case based on the results. Both the coefficients for salt and sugar

were large in magnitude and their confidence intervals were not symmetric. A thermodynamic

equilibrium condition was not considered. Despite an initial increase in melting rate, a

melting/refreezing equilibrium is eventually established between the salt water and the ice,

though at a lower temperature. Ambient water temperature around the cube drops to a level

where the rate of melting equals that of melted water refreezing back onto the cube (Senese

2010). This equilibrium is obtained at a solution temperature below that of pure water, and is

caused by a low chemical potential difference between the ice and the water solution as seen in

Figure 6 (Senese 2010). The result is an equilibrium freezing temperature below that of the ice

in the cube, which creates an incubating effect that increases the melting time of the cube

(Senese 2010).

0 5 10 15

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

Series resid(model)

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Figure 6. Relationship between chemical potential and melting temperature (Senese 2010)

Temperature had the greatest influence on the melting time based on both the magnitude of the

linear model coefficient and the unsymmetrical confidence interval not spanning 0. This is due

to a greater temperature gradient between the ice and the surrounding water resulting in an

accelerated heat loss from the cube, leading to fast melting times.

The pot & operator interaction was the highest of all the interaction terms. This is a surprising

result since it is saying that the results are different when one operator uses one pot versus

another operator using another pot. Independently, the coefficient for the pot factor is quite low

and insignificant when compared to the other factors. Furthermore, the confidence interval for

the pot spans 0 and is symmetric. This also shows that the pot is an insignificant factor. For the

operator, the linear model coefficient is not large in magnitude when compared to other

coefficients and the confidence interval is not as unsymmetrical as other confidence intervals.

Therefore, this is a really surprising result. A possible cause for this could be an unaccounted

disturbance such as convection heat transfer between the pot and heat emitting items other than

the stove element. This result is worth future investigation.

It was expected that the water type of water used (distilled water versus tap water) would have an

effect on the melting time. This was not the case. The confidence interval for water was

symmetric and the linear model coefficient was small in magnitude. The low effect on the

melting time could be due to the fact that cold water was used in the pot. As mentioned earlier,

cold tap water has fewer impurities than hot tap water. Perhaps, there were not enough

impurities in the cold tap water when compared to the distilled water to cause an effect.

7. Conclusions

The greatest effect on the melting rate of ice is temperature. Increasing the temperature

considerably increased the melting rate of ice. The next major effect on the melting rate of ice

was impurities. Unexpectedly, impurities had the opposite effect of what was expected.

Impurities slowed down the melting rate of ice rather than increasing the melting rate of ice.

Lastly, the next main effect on the melting rate of ice was the interaction between the type of pot

and the operator. The reason for this effect is unknown and possibly due to an unaccounted for

disturbance. Further areas of research to look into are: the interaction between the pot and the

operator and using salt ice cubes (ice cubes with salt in them) or sugar ice cubes rather than plain

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ice cubes in the experiment. The results of this experiment have applications with cooling

objects in ice water baths. For example, if cooling a beer in an ice water bath it would be

beneficial to keep the bath away from heat and to add impurities such as salt to the water. This

would help the ice last longer.

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8. Bibliography

Senese, Fred. “How can freezing point depression be explained in terms of free energies?”

General Chemistry Online! 2010.

http://antoine.frostburg.edu/chem/senese/101/solutions/faq/thermo-explanation-of-

freezingpoint-depression.shtml (accessed March 28, 2011).

Topper, Robert. “Ice Melting Principles.” Ask A Scientist. 2011.

http://www.newton.dep.anl.gov/askasci/chem99/chem99504.htm (accessed March 27,

2011).

Willis, Bill. “Salt and the Freezing Point of Water.” Worsley School. 2011.

http://www.worsleyschool.net/science/files/saltandfreezing/ofwater.html (accessed

March 27, 2011).

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9. Appendix A: R Code #List experiments as a fraction -> need to install FrF2 package

fract = FrF2(nruns = 32, nfactors = 6)

#Make experiments into two blocks -> need to install AlgDesign package

block = optBlock(~.,fract,c(16,16))

#Read list of experiments with results into R

results = read.csv('C:/Users/Alexander/Documents/School/Level4/Term2/ChemEng

4C03/ChemEng 4C03DOE/experiments.csv',sep = "\t")

attach(results)

#Create linear model

one = matrix(data = 1, nrow = 32, ncol = 1)

X = cbind(one, Salt, Sugar, Temperature, Pot, Operator, Water)

# A = Salt, B = Sugar, C = Temperature, D = Pot, E = Operator, F = Water

#Two factor interactions

X = cbind(X, X[,2] * X[,3]) #Salt * Sugar G = AB

X = cbind(X, X[,2] * X[,4]) #Salt * Temperature H= AC

X = cbind(X, X[,2] * X[,5]) #Salt * Pot I = AD

X = cbind(X, X[,2] * X[,6]) #Salt * Operator J = AE

X = cbind(X, X[,2] * X[,7]) #Salt * Water K = AF

X = cbind(X, X[,3] * X[,4]) #Sugar * Temperature L = BC

X = cbind(X, X[,3] * X[,5]) #Sugar * Pot M = BD

X = cbind(X, X[,3] * X[,6]) #Sugar * Operator N = BE

X = cbind(X, X[,3] * X[,7]) #Sugar * Water O = BF

X = cbind(X, X[,4] * X[,5]) #Temperature * Pot P = CD

X = cbind(X, X[,4] * X[,6]) #Temperature * Operator Q = CE

X = cbind(X, X[,4] * X[,7]) #Temperature * Water R = CF

X = cbind(X, X[,5] * X[,6]) #Pot * Operator S = DE

X = cbind(X, X[,5] * X[,7]) #Pot * Water T = DF

X = cbind(X, X[,6] * X[,7]) #Operator * Water U = EF

t(X) %*% X #Verify X is orthoganol

b = solve(t(X) %*% X) %*% t(X) %*% MeltingTime

model = lm(MeltingTime ~ Salt + Sugar + Temperature + Pot + Operator + Water)

summary(model)

#Make plots

library(car)

#QQ-Plot

qq.plot(resid(model), id.method = "identify")

hist(resid(model), breaks = 1)

#Time series residual plot

plot(resid(model))

abline(h = 0)

#Autocorrelation plot

acf(resid(model))

#Confidence Interval

confidenceinterval = confint(model)

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confidenceinterval

#Pareto plot

labels = c('A', 'B', 'C', 'D', 'E', 'F', 'AB', 'AC', 'AD', 'AE', 'AF', 'BC',

'BD', 'BE', 'BF', 'CD', 'CE', 'CF', 'DE', 'DF', 'EF')

N = length(b)

b.mod = abs(b[2:N]) # ignore intercept

idx = order(b.mod) # what is the sorted order?

b.mod = b.mod[idx]

labels.mod = labels[idx] # sort the labels in the same order as b.mod

library(lattice)

barchart(as.matrix(b.mod), ylab = "Effect", xlab="Magnitude of effect",

scales=list(y=list(labels=labels.mod)), col=0)

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10. Appendix B: Raw Data

The raw data collected from the experiments is presented in the following figures.

Figure 7. Operator 1 raw data

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Figure 8. Operator 2 raw data