Coffee Machine Analysis – Failure Rate Data Approach

Final Report

IE 540:585 Reliability Engineering I

Submitted by:

Omar Masood, Bilal Al Mula Abd, Bao Nguyen, Lei Xiao

Rutgers University

Industrial and System Engineering

Table Content

1. Introduction .........................................................................................................................1

2. Objectives ...........................................................................................................................2

3. System Description and reliability diagram..........................................................................2

3.1 Safety Features ..................................................................................................................3

3.2 Reliability Block diagram ..................................................................................................4

4. Methodology .......................................................................................................................4

4.1. Executive summary of methodologies ..............................................................................4

4.2. Failure Rate Based on Key Components Failure Rate .......................................................6

4.2.1 Methodology Approach for Electric Devices...............................................................6

4.2.2. Power Cord and Plug .................................................................................................7

4.2.3. Transformer ............................................................................................................. 10

4.2.4. Switch ..................................................................................................................... 11

4.2.5. Tube clogging by Calcium Carbonate ...................................................................... 13

4.2.6. The Principle Failure Rate ....................................................................................... 14

4.3. Failure Rate Estimation Based on Statistic ...................................................................... 14

4.4. Failure Rate Estimation Based on Simulated Data .......................................................... 17

4.5 Fitting the Data to Produce a Reliability Model ............................................................... 19

5. Data Collection Description .............................................................................................. 20

6. How to Extend Coffee Machine’s Lifetime ........................................................................ 21

6.1. Make Space for the machine ........................................................................................... 21

6.2. Follow Manufacturer Instructions ................................................................................... 21

6.3. Regular Cleansing Cycles ............................................................................................... 21

6.4. Replace Worn or Broken Parts ASAP ............................................................................. 21

6.5. Disconnect From the Power Source When not In Used ................................................... 21

6.6. Using the Filtered or Mineral Water Instead of Tap Water .............................................. 22

6.7. Consider Use Time Operation and Time Clock Device ................................................... 22

6.8. Power Cord and Plug Recommendation .......................................................................... 22

7. Conclusion ............................................................................................................................ 22

Appendix .................................................................................................................................. 23

References: ............................................................................................................................... 26

1

1. Introduction

Coffee is a very popular beverage all around the world. In fact, coffee is the second most

traded commodity worldwide. When evaluating the benefits of coffee, it becomes obvious why

coffee is so popular. It has been proven that coffee helps you live a longer and healthier lifestyle,

as it contains plenty of anti-oxidants and nutrients needed to survive (B2, B5, K, and Mg).

Coffee helps burn fat and increases metabolism by 3-11%, while improving workout

performance by increasing adrenaline and releasing fatty acids. It prevents Alzheimer’s and

Dementia by 65% and cirrhosis of the liver by 80%. With all these benefits, it is no surprise that

18.5 million drip coffee machines were purchased in the United States in 2010.

Coffee was first cultivated in Ethiopia in the sixth century A.D. The coffee berries were

consumed whole, or a wine was made out of the fermented fruits. Coffee, made from ground,

roasted beans, dates to the thirteenth century, and by the fifteenth century, coffee was popular all

across the Islamic world. The drink was introduced to Europe around 1615. The ancient method

of preparing coffee was to boil the crushed roasted beans in water until the liquid reached the

desired strength. The typical coffee pot was a long-handled brass pot with a narrow throat. This

kind of pot is still used throughout the Arab world, and is known in the West as a Turkish coffee

pot.

In England and America, boiling coffee in a sauce pan was for a long time the standard

method. Sometimes the coffee was boiled for several hours; other classic recipes called for

additions to the pot such as egg white, salt, and even mustard.

More sophisticated methods of brewing coffee evolved in France. The coffee bag, similar to

the familiar tea bag, appeared in France in 1711. Ground coffee was placed in a cloth bag, the

bag into a pot, and boiling water poured on top. Nearly a hundred years later, Jean Baptiste de

Belloy, who was Archbishop of Paris, invented a three-part drip coffee pot. The top part of the

pot held inside it a filter section made of perforated metal or china. Boiling water was poured

through the filter section, and it slowly dripped down to fill the pot below. The percolator was

invented in 1825. In a percolator, the pot full of water is placed directly on the stove burner.

When the water boils, it condenses in the top of the pot, and then drips through a strainer basket

filled with coffee. The Melitta filter, a plastic cone with several openings in the bottom, that

2

holds a paper filter of finely ground coffee, appeared around 1910, as did the glass Silex, an

hourglass-shaped filter pot.

The automatic drip coffee maker operates on the same principle as the Melitta and Silex, by

dripping boiled water through finely ground coffee in a paper filter. This machine debuted in the

United States in 1972 as the well-known Mr. Coffee. Mr. Coffee was an immediate success, and

popularized the automatic drip method. As of 1996, some 73% of American households report

owning an automatic drip coffee maker. Some models have timing features, so that they can be

pre-filled at night to make coffee at dawn. Other units have a temporary shut-off function, so the

carafe can be removed from the warmer plate while the coffee is filtering. Others pulse the water

over the filter at intervals, for a slower drip and more concentrated brew.

As an appliance used daily, the coffee maker may fail as its usage is increased, which can be

annoying and frustrating to the user. Many reasons can result in failure, for example the

components can wear heating element, power cord, etc. Or perhaps, the coffee will fail because

of damages caused by the user. While the coffee maker is a high reliable appliance, online

resources that will be discussed later report its average lifetime be approximately 6 years. In this

report, the main objective is to determine the reliability and safety of coffee makers and provide

suggestions on how to use coffee makers safely, and how to extend the lifespan of a coffee

maker.

2. Objectives

The main objective of this study is to determine the reliability and safety of coffee makers

utilizing the FAILURE RATE DATA APPROACH (λ). As coffee is such a valued commodity

worldwide, helping understand the reliability of coffee makers is quite important. Through the

use of the failure rate data approach, this study will determine the reliability of these machines,

discuss safety related issues, and recommend methods to prolong the life of a coffee machine.

3. System Description and reliability diagram

Like most kitchen appliances, coffee makers eventually fail over time. In order to analyze

reliability of the machine, the way the machine works must first be understood. In an automatic

drip coffee maker, a measured amount of cold water is poured into a reservoir. Inside the

reservoir, a heating element heats the water to boiling. The steam rises through a tube and

condenses. The condensed water is distributed over the ground coffee in the filter through a

device like a shower head. The water flows through the filter, infusing with the coffee, and falls

3

into a carafe. The carafe sits on a metal plate which has another heating element inside it. This

keeps the coffee warm. Simple research indicates that the key coffee machine components that

fail over time are the power cord/switch, heating element, inner tubes, heat sensors and the one-

way valve.

3.1 Safety Features

Overheating is the most dangerous cause of burning and explosion of the coffee maker. This

problem is not only a threat to machine damage but also a potential cause of fires. Because of the

serious risk caused by overheating, the overheating hazard protection system is the most

important part of coffee machine. Every coffee maker company want to make sure this

protection system will never fail due to the threat it may cause. There are three grades of

protection, as shown in figure 3.1.

The grade I protection is made up of two parallel thermal resistances. When the

temperature of the heater is too high, the thermal resistances will be activated and the heater is

restricted from melting.

The grade II protection is a thermostat. The thermostat is controlled by electricity and this

device is an important part in any thermal element. It ensures the safety of the heating element.

When the heater over-heats, the thermostat cuts off the power supply and it will re-connect after

the temperature cools down to normal.

The final grade (III) protection is a thermo cut off. This is the highest level of safety

protection. If the first two levels fail to activate when the temperature of the heater goes over the

threshold, the thermal cut off will sever the entire power supply for the machine. When the

thermal cut off activated, the coffee maker cannot reconnect to the power supply until the thermo

cut off is replaced.

Fig 3.1: Heating Protection System.

4

3.2 Reliability Block diagram

In order to understand the coffee machine system, a block diagram of all the components and

how they interact was created and is shown in figure 3.2 below.

Fig 3.2: Reliability Block Diagram of Coffee Machine Operation

4. Methodology

4.1. Executive summary of methodologies

The main objective for this project is to determine the reliability of a coffee maker using the

failure rate data approach. In order to do so, failure rate data must be collected. However, failure

rate data and specifically, the coffee machine failure data is hard to be collected in practice.

Online reviews by customers, along with very generic failure rate data were found online. In

order to overcome the shortcoming of data, four-methods were developed to estimate how

reliable the system is.

1. For the first method, the failure rate of the system is determined according to the failure

rate of each component. As illustrated in the reliability block diagram from section 4, all

of the coffee machine components are connected in series. In other words, the systems

5

remain functioning when all components function. Conversely, if a single component

fails, the entire machine fails. Therefore, the reliability of this series system as follow:

1

n

i

i

R R

(1)

Where n is the number of key components. Moreover, the failure principle in a series

system is equal the sum of the components failure rates:

1

n

i

i

(2)

According to the equation (2), the failure rate of the system is the sum of the failure rates

of each component. The following assumptions are considered in this methodology:

Assumption 1: All components function independently.

Assumption 2: The failure rate is subject to an exponential distribution.

Assumption 3: No distinction is made between complete failure and drift failures.

2. The second method is uses statistics of real values to analyze the reliability/failure rate of

the whole system. After collecting data from an online customer survey found online, the

hazard/failure rate is developed. Using this method, the following steps were performed.

Step 1. The time intervals were determined.

Step 2. The number of failures in the time interval were collected.

Step 3. The failure density, failure rate and reliability were calculated.

3. The third method utilized an artificial neural network (ANN) to build the reliability

model. As the ANN method requires a great deal of data, it is generated using online

information showing average coffee machine life. The ANN process used is as follows.

Step 1. The time intervals were determined.

Step 2. The number of failure in each time interval were collected.

Step 3. The network and corresponding parameters was constructed, along with

inputs and outputs.

Step 4: The reliability was simulated using ANN.

4. The fourth method analyzed reliability by fitting the data found in method 2 and 3 using

the toolbox of Matlab. Using this approach, the reliability/failure rate of the whole system

was analyzed. In this method, the reliability of the coffee maker was assumed to be

6

subject to a certain distribution. Then the data-fitting toolbox of Matlab was used to

estimate the corresponding parameters automatically.

4.2. Failure Rate Based on Key Components Failure Rate

Some preliminary research indicates that there are twenty to twenty-five main components

that need to function in order for the coffee machine operate. By calculating the failure rate of

certain key components, the principle failure rate can be estimated based on the proportion of

these key components. The main components include the power cord, plug, transformer, switch

and tubes.

4.2.1 Methodology Approach for Electric Devices

The failure rate of the system is calculated by summing up the failure rates of each

component in each category (based on probability theory). The following models assume that the

component failure rate under reference or operating conditions is constant. According to existing

reference, it is justified to use a constant failure rate for each component. This may take the form

of analyses of likely failure mechanisms, related failure distributions, etc. In this part, the failure

rate for the electric component is estimated by two methods:

1. Failure rate prediction at reference conditions (parts count).

The failure rate for the equipment under reference conditions is calculated as follows:

_ e

1

( )n

Part count r f i

i

(1)

Where, ref is the failure rate under reference conditions, and n is the number of

components.

The reference conditions adopted are typical for the majority of applications of components

in equipment. It is assumed that the failure rate used under reference conditions is specific to the

component and includes the effects of complexity, technology of the casing, different

manufacturers and the manufacturing process etc.

2. Failure rate prediction at operating condition (part stress).

Components in equipment may not always operate under the reference conditions. In such

cases, the real operational conditions will result in failure rates different from those given for

reference conditions. Therefore, models for stress factors, by which failure rates under reference

conditions can be converted to values applying for operating conditions (actual ambient

7

temperature and actual electrical stress on the components), and vice versa, may be required. The

failure rate for equipment under operating conditions is calculated as follows:

_

1 1

( )mn

Part stress ref j i

i j

(2)

Where, ref is the failure rate under reference conditions, n is the number of components, m is

the number of factors.

In this project, the failure rate of components was calculated by using the method of part

stress manner using data from MIL-HDBK-217 F[7]. The failure rate estimation process consists

of the following steps:

Step 1. Define the equipment to be analyzed.

Step 2. Understand system by analyzing equipment structure.

Step 3. Determine operational conditions, including operating temperature and rated stress.

Step 4. Determine the actual electrical stresses for each component.

Step 5. Select the reference failure rate for each component from the database [7].

Step 6. In the case of a failure rate prediction at operating conditions calculate the failure

rate under operating conditions for each component using the relevant stress models

Step 7. Sum up the component failure rates.

Step 8. Document the results and the assumptions.

4.2.2. Power Cord and Plug

The connection of electronic equipment to the AC power supply is usually accomplished

using detachable connectors. The alternative of "hard-wiring" equipment to the building wiring

makes service and movement of equipment more costly and less convenient. Therefore, many

types of connectors exist. As a result, much confusion is generated as to what the various

connection types are, when they are used, and what they should look like. But, if not used

properly, power cords lead to electrical shock hazards, equipment damage, and fire hazards.

4.2.2.1. Types of Power Cord

Power Cords come in either two or three-wire types. Two-wire cords should be used to

operate small appliances. Three-wire cords are used for outdoor appliances and electric power

tools. The third wire on the cord is a ground. This type of cord should never be plugged into an

ungrounded electrical outlet. Only grounded cords shall be used with power tools. One

exception to this is if the power tool is double insulated.

8

Table 4.1: Power Cord Types and Rating

Power Cord Ampere Rating

Wire Size (Copper) Single Phase Two and Three Conductor Cords Three Phase Cords

16AWG 13 amps 10 amps

14AWG 18 amps 15 amps

12AWG 25 amps 20 amps

10AWG 30 amps 25 amps

8AWG 40 amps 35 amps

6AWG 55 amps 45 amps

4AWG 70 amps 60 amps

2AWG 95 amps 80 amps

4.2.2.2. Electrical Plug types

There are two different styles of AC connectors that exist in order to address different

wiring systems and to ensure user safety:

Fig 4.1: US Standard Type “A”

The US standard type "A" plug has two flat prongs, with or without holes, one slightly

larger than the other.

9

Fig 4.2: US Standard Type “B”

The US standard type "B" plugs has two flat prongs, with or without holes, one slightly

larger than the other and a third grounding pin/prong.

Table 5.2 Applications and Pins Count for Different Wiring Systems.

4.2.2.3. Common Power Cord Problems

There are many way that lead to damage of a power cord:

1. A surge typically measures less than 500V and lasts less than two seconds. A spike, by

definition, is much shorter in duration - less than one-thousandth of a second

(millisecond), but can measure into the thousands of volts. Either type of disturbance can

damage electronic equipment beyond practical repair. In addition to change in demand

for electricity, bad weather (lightning) and everyday electric utility company switching

and maintenance can produce damaging electrical surges on the power line.

2. Aging of insulating layer and open circuit of wires.

3. Water leaks cause short circuits.

4. Cutting the plug blades or grounding pin of an extension cord or appliance to plug it into

an ungrounded outlet.

10

4.2.2.4. Failure Rate of Power Cord:

The failure rate equation of power cord as follows:

60.007 3.2 2 4 3 0.5376 10 /p ref T K Q E hr (5)

Where, ref is the base or reference failure rate, T is temperature factor, K is mating /

un-mating factor,Q is quality factor, and E is environment factor

4.2.2.5. Failure Rate of Plug:

The failure rate equation of the plug is as follows:

60.00064 1.5 1 3 0.00288 10 /plug ref P Q E hr (6)

Where, refis base or reference failure rate, P is active pins factor, Q is quality factor,

and E is environment factor

4.2.3. Transformer

The power transformer is one of the key components in any electrical devices. This device

transfers energy between two or more circuit through electromagnetic induction. It is used to

either step up (raise), or step down (lower) electricity between circuits. In the coffee machine, the

transformer reduces the voltage/current from the outlet to a reasonable value in order to running

the machine.

4.2.3.1. Common Transformer Problems

There are number of factors which affect the life time expectancy of a power transformer.

According to the paper from National Power Systems Conference (1), the common factors

include:

1. Overloading, this occurs when the utility increases the load over time. As a result, the

capacity of the transformer is eventually exceeded, leading to failure.

2. Moisture, which failures caused by water leaking from the machine; the environment the

coffee maker is placed in, or human mistake.

4.2.3.2 Failure Rate of Transformer:

The transformer’ failure rate was calculated as follows:

60.0045 2.4 30 6 2.3328 10 /Transformer ref T Q E hr

(7)

11

Where, ref is base/reference failure rate, T is temperature factor, Q is quality factor,

and E is environment factor

4.2.4. Switch

4.2.4.1 Switch Types

Switch types commonly found on consumer devices include:

1. Rocker switch - 2 states. Switches between on and soft-off or on and hard-off. May be

movable to off by automatic means.

2. Rocker switch - 3 states, with on a momentary state. The intermediate state of the switch

is on or automatic off.

3. Push-button - 2 state, with a mechanically observable difference between the two states.

Can be a notebook lid switch.

4. Momentary contact switch — a button or slider. With only one stable state. Moving the

switch may cause a transition to the opposite state, or always to on.

Fig4.3 Different switch types

4.2.4.2 A Application for Switches

Some common applications are clear with the present symbols. Examples are devices

with:

A rocker switch in which off is zero power; it will be labeled with I and

A push-button 2 state switch in which off is zero power; it will be labeled

A push-button or momentary contact switch with non-zero power in off; it will be labeled

with

Fig 4.4

12

4.2.4.3 Common Switch Problems

Some switches are not good switches for these applications as they: raise ambiguities,

inconsistencies, and confusion. These can lead to annoyance, energy waste, and perhaps safety

concerns. The following are examples:

1. Soft-off. Some devices have a rocker switch that toggles between on and soft-off. When

this occurs on office equipment, it usually has I for on and for off. The problem with

this is that it identifies as meaning off, whereas when it is used on a power button,

people interpret it as meaning power on.

2. Multiple power switches. Other devices have two power switches: one which controls the

functional power state (for which the off power level is not important) and the other

which is used to switch the device to zero power. User manuals often call the latter a

“main power” switch. The question arises as to whether the icon labeling of the two

switches should make clear their relationship, or whether cues such as location are always

sufficient (e.g. the main power switch being on the back of the device near where the

power cord enters).Regardless, if the main power switch goes to zero power on or off, it

should have the I and symbols.

3. Unknown off power. In some contexts, the power consumption while off may not be

known or may change. This occurs in operating systems that may not know the power

status of the hardware they run on and so may not know which symbol to use. This also

can occur with devices that can be operated on battery or mains power; their status while

off may vary depending on whether the device is mains-connected, and also whether the

battery is present

4.2.4.4 Failure Rate of Switch

The failure rate of the switch can be calculated by the following equation:

1 2

6

( )

(0.086 0.089 2) 1 1.76 3 1.3939 10 /

Switch ref N ref cyc L E

hr

(8)

Where, refis the base or reference failure rate, N is the number of active contacts, L is

the load stress factor, E is the environment factor, and cycis the cycling factor.

13

4.2.5. Tube clogging by Calcium Carbonate

The tube clogging is the least concern among the three common failures in coffee maker.

This problem happens rarely and is due to a slow chemical reaction. In addition, many treatments

can be applied to the coffee machine, such as running malt vinegar or lemon juice, which easily

dissolves the mineral deposits.

4.2.5.1. Cause of Tube Clogging

Hard water has a high mineral content. This usually consists of high levels of metals ion

(calcium, and magnesium in the form of carbonates). Although, hard water is not generally

unhealthy for human consumption, there is still the potential for health problems to occur.

Furthermore, when hard water evaporates inside pipes or tube, it produces Limescale. Limescale

is a hard, off-white (99% Calcium Carbonate), chalky deposit that forms on the inner surface of

old pipes and tubes where hard water has evaporated.

5.2.5.2. Dangerous of Limescale

There are many problems caused by hardness in water. For housing, clogged pipes can

decrease the life of toilet flushing units by 70% and water taps by 40% (1). Limescale can serve

as a medium for bacterial growth, causing nappy rash, and minor skin irritation. For industries,

coating of limescale on heating element can make it up to 12% less effective and waste energy

according to British Water reports (2). These are serious problems for any industry.

5.2.5.3. Methodology

The objective is to evaluate the failure rate of tube clogging. By studying the kinetics of

calcium carbonate scale deposition on heat-transfer surfaces, we can estimate the length of time

until the heating tube develops a 1.6 millimeter thick of Limescale wall. With 1.6 millimeter

Limescale deposits on the inner surface of the heating tube, the heating element is considered to

fail because of the 25% increase to the electrical bill and 12% less effective functioning of the

machine. The following were assumptions made for this analysis:

Assumption 1: The density of calcium ion contain in hard water is 140 mg/l (According

to the United State Geological Survey [4]).

Assumption 2: For the inner surface area of the heating tube, the structure was simulated

using SolidWork (3D design software). Using the measuring-tool, the inner surface area and

volume of the heating tube were estimated. The inner area surface was found to be 73.35 cm3.

Assumption 3: Thermal conductivity of scale is: ks = 1.45 kcal/ hr/meter oC [8]

14

The scale-growth rate (in weight per unit time per unit area), as follow:

0

 . .bs s

s

d T Tp Akdxw p

dt q dt

(9)

Where, ps is density of scale (120mg/l). A is surface area of the heating tube, ks is the

thermal conductivity of scale, q is the heating flow per unit time, d(To - Tb)/dt is the slope of

straight line representing the variation of temperature difference (To - Tb) with time.

Heating flow in the above equation can be calculated by following equation:

 cq h AdT (10)

Where, q is the heat transferred per unit time (W). A is the heat transfer area of the

surface (m2). hc is the convective heat transfer coefficient of the process (W/(m

2K)) or W/ (m

2

0C)). dT is the temperature difference between the surface and the bulk fluid (K or

oC).

According to the mechanism of calcium carbonate scale deposition on heat-transfer

surfaces, the nucleation of CaCO3 deposits on metallic surface is not significantly different from

the subsequent growth on CaCO3 crystals. Therefore, the kinetics of this chemical reaction is

considered as a constant rate. Because the chemical reaction rate is unchanged, the failure rate of

tube clogging by a chemical reaction is constant. The failure rate can be estimated by 1/MTTF.

Assuming that a 90% volume of calcium carbonate in the heating tube is the signal of failure and

based on the two aforementioned equations, the estimator for failure rate of tube clogging is

51.3896 10 / hr . The calculation is explained further in the appendix by Matlab.

4.2.6. The Principle Failure Rate

Applying equation (2), the summation failure rate of these main components (power cord,

plug, transformer, switch and heating tube) is 51.8163 10 / hr . However, as previously stated,

there are approximately twenty-five different main components both electrical and non-electrical

inside the coffee machine. As such, these five key components only represent 20% of the entire

system failure rate. Therefore, after incorporating the remaining 80%, the system failure rate is

estimated to be 59.0816 10 / hr . If we assume the average operation time is 5 hours per day, the

mean time to failure is easily calculated to be approximately 7.52 years.

4.3. Failure Rate Estimation Based on Statistic

The statistics method is data-driven, which means analyzing the failure rate using lots of

samples. The most important part of a data-driven method is collecting data and analyzing the

potential properties of the data.

15

The lifetime data was found by investigating the average lifespan of coffee makes from a

Canadian website [2]. According to the website, poll results for the average lifespan of drip

coffeemakers are given as follows.

Fig 4.5 Average Lifespan of Drip Coffee Maker

From Fig 4.5, the time interval is divided in a large range. More details about the lifespan

over 3 years should be known. Therefore, each customer’s comment is reviewed. After this

review, data is retrieved indicating specific coffee machine lifetimes between 3 and 20 years.

With this data, the failure rate (f(t)), hazard rate (h(t)), reliability (R(t)) and cumulative

distribution function of failure (F(t)) can be calculated by using the following equations:

f

o

n tf t

n t

(3)

f

s

n th t

n t t

(4)

s

o

f t n tR t

h t n (5)

1F t R t (6)

Where, nf(t) is the number of failed components. no is the number of identical components

which are subjected to the design operating condition test. ns(t) is the number of surviving

components at the beginning of the period ∆t. The results are tabulated in Table 4.3.

Table 4.3 f(t), h(t), R(t) and F(t) for Coffee Machine Using Canadian Data

Time interval

(years)

Failures in the

interval

Failure density

f(t)

Hazard rate

h(t)

Reliability

R(t)

Unreliability

F(t)

0-1 3 0.1579 3/19=0.1579 1 0

1-2 2 0.1053 2/16=0.1250 0.8424 0.1576

2-3 2 0.1053 2/14=0.1429 0.7369 0.2631

3-4 1 0.0526 1/12=0.8333 0.0631 0.9369

4-5 2 0.1053 2/11=0.1667 0.6317 0.3683

16

5-9 0 0 0/9=0 - -

9-10 6 0.3158 6/9=0.6667 0.4737 0.5263

10-11 0 0 0/9=0 - -

11-12 1 0.0526 1/3=0.3333 0.1578 0.8422

12-17 0 0 0/9=0 - -

17-18 1 0.0526 1/2=0.5 0.1052 0.8948

18-19 0 0 0/9=0 - -

19-20 1 0.0526 1/1=1 0.0526 0.9474

Total 19

As can be seen from Table 4.3, there are some intervals with zero failures. For these

specific intervals, the corresponding failure rates, and hazards rate are 0, and the reliability is

meaningless (0/0). Therefore, the data was regrouped using five year intervals and displayed in

Table 4.4.

Table 4.4 Results of f(t), h(t) and R(t) Using Five Year Intervals

Upper bound

(years)

Number failing Number surviving Reliability Failure

density

Hazard rate

0 0 19 1.0000 0.1053 0.1053

5 10 9 0.4737 0.0632 0.1333

10 6 3 0.1578 0.0105 0.0667

15 1 2 0.1053 0.0211 0.2000

20 2 0 0

total 19

The reliability R(t), failure density f(t), hazard rate h(t), MTTF and the sample variance is

estimated as follows.

1,2, ,inR t i k

n (7)

1

1

1

for i i

i i

i i

R t R tf t t t t

t t

(8)

1for i i

f tt t t t

R t (9)

17

1

1 10 0

0

02

ki i i i

i i

i

n n t tMTTF t t t n n

n

(10)

Where n is the number of units at risk at the start of the test and ni is the number of units

having, survived at ordered time ti respectively.

Finally, the MTTF is calculated to be 6.1842. Also, another resource was found online that

displays the lifetime of coffee machines [5]. Typical lifetime of coffee machines in the United

States, by product type and in years is illustrated in Fig 4.6.

Fig 4.6. Typical Lifetime of Coffee Machines in the United States, by Product Type (in years)

When comparing the MTTF of the coffee maker calculated from Canadian website data with this

data is very clear that the values are close. This is a strong indication that the estimate is accurate

4.4. Failure Rate Estimation Based on Simulated Data

Due to the small amount of failure samples utilized simulation data was simulated to develop

another failure rate and reliability model. According to the statistics from the US website and the

Canadian website, the mean lifetime of a coffee maker is approximately 6 years. In addition, the

known maximum lifetime is 20 years. Therefore, it is assumed that a chi-square distribution is

more appropriate to simulate the lifetime of a coffee maker. As such, one hundred samples of

data were simulated to build the lifetime samples. The simulated mean lifetime was 6.1473

years, the maximum lifespan was 23.4526 years, and the minimum lifespan was 0.8955 years.

The lifespan distribution is as follows.

18

Time (Years)

Fig 4.7 Histogram of Coffee Machine Lifespan Using Simulated Data

Next, the lifespan was divided into 0.5 time intervals (year), and the frequency, reliability,

and corresponding unreliability was calculated.

Time (Years)

Fig 4.8 Unreliability of Coffee Machine Using Simulated Data

The failure rate model for a coffee maker is built using an artificial neural network (ANN)

based on [3]. The basic settings and parameters of ANNs are given as follows: there is one input

layer and one output layer, the input of the network is the operating time of the coffee maker, and

the output is the corresponding reliability. There is one neuron in both the input and output

0 5 10 15 20 250

2

4

6

8

10

12

14

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fre

quency

F

ail

ure

F(t

)

19

layers. There is one hidden layer, which has ten neurons. “Tansig” is the activiation function of

the neurons in the hidden layer, “purelin” is of the output layer. The weight adjustment is based

on the Levenberg-Marquardt method because of its fast convergency. The maximum iteration is

10000, and the error performance goal is 1e-10. The error evaluation is calculated based on

Equation (14). The comparison of the simulated reliability and real reliability is given as follows.

2

error pr rr (11)

Where, pr is the predicted reliability, and rr is the real reliability.

Time (Years)

Fig 4.9 Comparison of Coffee Machine Reliability Between Predicted Reliability Based on ANN and

Real Reliability

The final error is 5.65442e-006 based on Equation (14).

4.5 Fitting the Data to Produce a Reliability Model

The Weibull distribution is extensively used to estimate system reliability. In addition,

many different types of electronic appliances exhibit exponential degradation. Therefore, data

fitting was considered when building a reliability model. The estimated results of exponential

distribution are listed as follows: 0.1456tR t e with 95% confidence, 0.1299 0.1612 ,

and fitted error is 0.45. Similarly, the estimated results using a Weibull distribution are as

follows:

1.933

7.252

t

R t e

. with 95% confidence, the minimum and maximum estimated

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

real reliability

predicted reliability

Reli

abil

ity R

(t)

20

parameters are 1.857 2.009 and 7.149 7.355 , and the estimated error is 0.01511.

The performances of the fitted models are illustrated in Figure 4.10 below.

Time (Years)

Fig 4.10 Fitting by using exponential distribution and Weibull distribution

5. Data Collection Description

Four different approaches were used to estimate the reliability of a coffee maker. They can

classified in two ways, one is estimation based on key components, while the other three

methods can be regarded as data-driven methods.

For the main component failure rate technique, the failure rates of the power cord and heating

element were retrieved from the Military hand book. The failure rate of the heating tube (caused

by clogging) was estimated by the kinetic reaction of calcium carbonate. In order to use this data,

the scale growth rate equation was applied to convert the data to a coffee machine. To obtain

thses parameter, such as the density of calcium ion in tap water, or the temperature of the heating

element function, estimation were made based on product in website databases and instruction

handbooks form many coffee maker companies.

For the data-driven approach, the failure rate of the coffee machine was constructed using data

from a website survey. In addition, to retrieve more details about the lifespan, data collected

from the customer’ review was utilized. Finally, oo enlarge the lifetime samples, simulated data

was generated based on real-world data.

Reli

abil

ity R

(t)

21

6. How to Extend Coffee Machine’s Lifetime

As one of the most used small appliance in the home, coffee maker deserves tender loving

care to keep them in top form. It is recommended that user apply these top eight tips in order to

extend coffee machine service.

6.1. Make Space for the machine

Because the coffee maker is one of the smallest kitchen appliances, people tend to take it for

granted. However coffee machines generate a considerable amount of heat when it brews coffee.

For safety reasons and to allow the machine to cool properly, it is recommended that the coffee

maker have enough space between it and the next small kitchen appliance on the counter top[11].

6.2. Follow Manufacturer Instructions

Although coffee makers tend to look alike that each make and model was designed to

provide a few unique functions. That is why it is important to refer to the manufacturer’s manual

regularly to ensure that you get optimum performance from your coffee maker. Following

instruction from the manual also prevents misuse that can lead to damage.

6.3. Regular Cleansing Cycles

It is not enough to simply rinse out the carafe at the end of the day. To ensure each batch of

coffee tastes freshly made, it’s highly recommended to put your coffee maker through regular

cleansing cycles. Depending on how often you brew coffee, a once-a-week thorough cleanse is

usually ideal.

This cleanse should consist of brewing a water and vinegar solution or water and

dishwashing detergent mixture, similar to eh manner in which coffee is brewed. This is will

sanitize the machine, but, more importantly, prevent clogging of the heating tubes. Conversely,

6.4. Replace Worn or Broken Parts ASAP

When thoroughly cleaning the coffee maker, one should also check for parts that may be

showing signs of wearing out or breaking down. If a slight crack shows up on the carafe or

coffee pot, replace it immediately. Worn out power cords and chipped plastic parts, like the filter

basket, should also be right away.

6.5. Disconnect From the Power Source When not In Used

If one is away from home for a long period, such as an out-of-town trip, to the coffee maker

should be disconnected from the power source. Most house fires were found to be caused by

22

appliances that have been left plugged to the electrical outlet over extended periods even when

not in use.

6.6. Using the Filtered or Mineral Water Instead of Tap Water

In order to avoid the Limescale process, water should be filtered before feeding into the

reservoir of the machine. Using distilled or mineralized water as opposed to tap water will

prevent clogging of the heating tubes.

6.7. Consider Use Time Operation and Time Clock Device

From start to finish, most coffee machines take approximately 5-15 minutes to produce a cup

of coffee. However, most machines operate much longer than this as they are left unattended

until the user decides to drink the coffee. Using a timed device automatically shuts off the

machine after a predetermined time, thus prolonging coffee machine life.

6.8. Power Cord and Plug Recommendation

Manufacturers should use type ‘B’ US standard plug with a grounding pin when producing

coffee machine. This will certainly minimize, if not, alleviate safety issues caused by type ‘A’

plugs. Moreover, the power should be plugged directly into the outlet which can provide stable

current/voltage. These two things can extend your life-time of the coffee maker.

7. Conclusion

As the coffee machine is one of the most utilized appliances in a household, knowing the life

time and methods to increase the age, would be quite useful, Through the use of four varying

methodologies, the lifetime of a coffee machines has been shown to range from 6-7.52 years. As

the difference in time between all four methods is not significant, it is believed that this range is

highly accurate. Furthermore, steps have been outlined that will almost certainly prolong, coffee

machine life much longer than this aforementioned range.

23

Appendix

Appendix A - MATLAB Code for Tube Clogging

%%Fixed variable %%Density of scale (density of calcium ion in hradwater) Kg/cm^3 Ps =0.12; %%Area of inner aluminum tube surface A = 0.007335; %%thermal conductivity of scale Ks = 1.45; %%heating flow equation q =2000*0.007335*(373-294); %% Slope of straight line representing the variation of temp differ Tmax = 100; Tmin = 21; T1= 0.5/60; To = 0; dT =(Tmax -Tmin)/(T1-To); %% The scale-groeth rate (weight per unit time per unit area) (kg/hr.m^2) w=(Ps*A*Ks*dT)/q %% kg/hr.m^2 convert to g/hr.cm^2 w1 = w*1*10^3/1*10^-4 %% Assume coffee machine operate 10 mins/day and Area (73.35cm^3) w2 = w1*(1/6)*73.35 %% Volume of calciuum carbonate / day 1gram = 0.369 cm^3 V= w2*0.369 %% Total voume inside of the heating tube 18.305 cm^3 %% How many date untill the tube get 90% occupied day = 16.4745/V year = day/365 Failure_rate = 1/(day*24)

24

Appendix B – MATLAB Code for Simulation

%% plot hist clc; clear all %% generate chi-square distribution failure samples. close all clear clc x=chi2rnd(6,100,1); %simulated lifetime hist(x,25); mean_value=mean(x) max_value=max(x) min_value=min(x) max_life=ceil(max_value); %% statistics failure_number=[]; if max_life-fix(max_life)>0.5 m=ceil(max_life); else m=fix(max_life)+0.5; end axisx=1:0.5:m; for i=1:0.5:m y=i-1:0.5:i-0.5; n=histc(x,y); failure_number_temp=n(1); failure_number=[failure_number;failure_number_temp]; end reliability=(length(x)-cumsum(failure_number))./length(x); failure_rate_cdf=1-reliability; figure plot(failure_rate_cdf) figure plot(reliability) %% ANN part for reliability input=(1:0.5:m)/max_life/10; output=reliability'; net = newff(input,output,10,{'tansig', 'purelin'}, 'trainlm'); net.trainParam.show=1; net.trainparam.epochs=10000; net.trainparam.goal=1e-10; net=train(net,input,output); reliability_sim=sim(net,input); for i=1:length(reliability_sim) if reliability_sim(i)>1 reliability_sim(i)=1; end if reliability_sim(i)<0 reliability_sim(i)=0; end end error=1/2*(sum(reliability_sim-reliability').^2) hold on plot(reliability_sim,'r-') legend('real reliability','predicted reliability')

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Appendix C- Image of Heating Tube and Simulated Image

3D model of heating aluminum tube by SolidWork

26

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