**1. Introduction**

224 Energy Efficiency – The Innovative Ways for Smart Energy, the Future Towards Modern Utilities

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1952—2000. Economic Research Journal, 2004. 10: p. 35-44.

Reliability refers to the ability of system or component to perform a required function under stated environmental and operational conditions for a specified period of time. Traditionally, the reliability over the product life can be illustrated by a bathtub curve that has three regions: a decreasing rate of failure, a constant rate of failure, and an increasing rate of failure, as shown in Figure 1(a). As the reliability of a product (or part) improves, failure of the part becomes less frequent in the field. The bathtub curve may change into a straight line with the slope angle β. In a straight line there are two variables to be measured: product life *LB* (or mean time between failures) and failure rate λ, as shown in Eq. (1):

© 2012 Woo et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

$$\mathcal{R}\left(\mathcal{L}\_{\mathcal{B}}\right) = e^{-\mathcal{\lambda}L\_{\mathcal{B}}} \equiv 1 - \mathcal{\lambda}L\_{\mathcal{B}} \tag{1}$$

The Reliability Design and Its Direct Effect on the Energy Efficiency 227

λ

The multi-unit refrigerator used as a case study for this method consists of a compressor, a drawer, a door, a cabinet, and other units. For the drawer, the *B1* life of the new design is targeted to be over 10 years with a yearly failure rate of 0.1%. The entire refrigerator's *BX* life can be obtained by summing up the failure rates of each refrigerator unit. The refrigerator's *B12* life with the new design is targeted to be over 10 years with a yearly failure rate of 1.2 %

No Units Market Data Design Conversion Expected Target *Bx* Life Based *Bx* Failure Rate *Bx* Life Failure Rate *Bx* Life 1 Compressor 0.34 5.3 New x5 1.70 0.10 10 B1.0 2 Door 0.35 5.1 Given x1 0.35 0.15 10 B1.5 3 Cabinet 0.25 4.8 Modified x2 0.50 0.10 10 B1.0 4 Drawer 0.20 6.0 New x2 0.40 0.10 10 B1.0 5 Heat exchanger 0.15 8.0 Given x1 0.15 0.10 10 B1.0 6 etc 0.50 12.0 Given x1 0.50 0.50 10 B6.0 Sum R-Set 1.79 7.4 - - 3.60 1.10 10 B12.0

**1.2. Analysis of the problems identified in field samples (loads analysis)** 

In the field, certain components in these refrigerators had been failing or making noise, causing consumers to replace their refrigerators. Data from the failed products in the field showed how common used the refrigerators under common usage conditions. Refrigerator reliability problems in the field occur when the parts cannot endure repetitive stresses due to internal or external forces over a specified period of time. The energy flow in a refrigerator (or other mechanical) system can generally be expressed as efforts and flows

**(or Parts) Effort,** *e(t)* **Flow,** *f(t)*

(draws, dispenser lever) Force component, *F(t)* Velocity component, *V(t)*

(PCB, condenser) Voltage, *V(t)* Current, *i(t)*

(door, cooling fan) Torque component, τ*(t)* Angular velocity component, *V(t)* Compressor Pressure difference, Δ*P(t)* Volume flow rate, *Q(t)*

For a mechanical system, the time-to-failure approach employs a generalized life model (LS

**1.1. Targeting the refrigerator** *BX* **life and failure rate** 

**Table 1.** Total parametric ALT plan of refrigerator

(Table 2) [15]. Thus, the stresses come from the efforts.

**Refrigerator Units**

Mechanical translation

Mechanical rotation

Electric

model) [16], such as:

**Table 2.** Effort and flow in the multi-port system

(Table 1) [19].

We can thus establish the reliability growth plan of parts with a constant failure rate.

A company generally designs its new products to (1) minimize initial failures, (2) reduce random failures during the expected product working period, and (3) lengthen product life. Such aims are met through the use of robust design techniques, including statistical design of experiment (SDE) and the Taguchi methods [1]. The Taguchi methods describe the robustness of a system for evaluation and design improvement, which is also known as quality engineering [2-3] or robust engineering [4]. Robust design processes include concept design, parameter design, and tolerance design [5]. Taguchi's robust design methods place a design in an optimum position where random "noise" does not cause failure, which then and helps in determining the proper design parameters [6].

However, for a simple mechanical structure, the Taguchi methods' robust design processes need to consider a large number of design parameters. They also have difficulty in predicting the product life, *LB* (or MTBF).

In this study we present a new method for the reliability design of mechanical systems. This new method takes into account the fact that products with missing or improper design parameters can result in recalls and loss of brand name value. Based on the analysis of a failed refrigerator drawer and handle systems, we demonstrated our new reliability design method. The new method uses ALT; the new concept of product life, *LB*; and sample size, as a novel means of determining proper design parameters [7-14].

#### **1.1. Targeting the refrigerator** *BX* **life and failure rate** λ

226 Energy Efficiency – The Innovative Ways for Smart Energy, the Future Towards Modern Utilities

(b) System life and failure rate consisting of unit #1, unit #2 and unit #3

A company generally designs its new products to (1) minimize initial failures, (2) reduce random failures during the expected product working period, and (3) lengthen product life. Such aims are met through the use of robust design techniques, including statistical design of experiment (SDE) and the Taguchi methods [1]. The Taguchi methods describe the robustness of a system for evaluation and design improvement, which is also known as quality engineering [2-3] or robust engineering [4]. Robust design processes include concept design, parameter design, and tolerance design [5]. Taguchi's robust design methods place a design in an optimum position where random "noise" does not cause failure, which then

However, for a simple mechanical structure, the Taguchi methods' robust design processes need to consider a large number of design parameters. They also have difficulty in

In this study we present a new method for the reliability design of mechanical systems. This new method takes into account the fact that products with missing or improper design parameters can result in recalls and loss of brand name value. Based on the analysis of a failed refrigerator drawer and handle systems, we demonstrated our new reliability design method. The new method uses ALT; the new concept of product life, *LB*; and sample size, as

*<sup>B</sup> e L* λ

λ

<sup>−</sup> = ≅− (1)

RL 1 ( ) <sup>B</sup> *BL*

We can thus establish the reliability growth plan of parts with a constant failure rate.

and helps in determining the proper design parameters [6].

a novel means of determining proper design parameters [7-14].

predicting the product life, *LB* (or MTBF).

**Figure 1.** System life and failure rate

The multi-unit refrigerator used as a case study for this method consists of a compressor, a drawer, a door, a cabinet, and other units. For the drawer, the *B1* life of the new design is targeted to be over 10 years with a yearly failure rate of 0.1%. The entire refrigerator's *BX* life can be obtained by summing up the failure rates of each refrigerator unit. The refrigerator's *B12* life with the new design is targeted to be over 10 years with a yearly failure rate of 1.2 % (Table 1) [19].



## **1.2. Analysis of the problems identified in field samples (loads analysis)**

In the field, certain components in these refrigerators had been failing or making noise, causing consumers to replace their refrigerators. Data from the failed products in the field showed how common used the refrigerators under common usage conditions. Refrigerator reliability problems in the field occur when the parts cannot endure repetitive stresses due to internal or external forces over a specified period of time. The energy flow in a refrigerator (or other mechanical) system can generally be expressed as efforts and flows (Table 2) [15]. Thus, the stresses come from the efforts.


**Table 2.** Effort and flow in the multi-port system

For a mechanical system, the time-to-failure approach employs a generalized life model (LS model) [16], such as:

$$T\_f = A \left(\text{S}\right)^{-n} \exp\frac{E\_a}{kT} = A \left(e\right)^{-n} \exp\frac{E\_a}{kT} \tag{2}$$

The Reliability Design and Its Direct Effect on the Energy Efficiency 229

**, with the improved designs** 

+ (9)

(10)

(11)

of cycles and the number of required test cycles can be obtained from Eq. (7). ALT equipment can then be conducted on the basis of load analysis. Using ALT we can find the

The parameter design criterion of the newly designed samples can be more than the target life of *BX* = 10 years. From the field data and from a sample under ALT with a corrective action plans, we can obtain the missing or improper parameters of parts and their levels in

With the improved design parameters, we can derive the expected *LBx* life of the final design

1 *<sup>B</sup>*

⋅ ⋅ ≅ ⋅

( )

**2. Case study: Reliability design of a refrigerator drawer and handle** 

*r*

≅ ⋅+⋅

<sup>1</sup> <sup>1</sup> *<sup>B</sup>*

*L n h AF*

Figure 2 shows a refrigerator with the newly designed drawer and handle system and its parts. In the field, the refrigerator drawer and handle system had been failing, causing consumers to replace their refrigerators (Figure 3). The specific causes of failures of the refrigerator drawers during operation were repetitive stress and/or the consumer improper usage. Field data indicated that the damaged products had structural design flaws, including sharp corner angles and weak ribs that resulted in stress risers in high stress areas. A consumer stores food in a refrigerator to have convenient access to fresh food. Putting food in the refrigerator drawer involves opening the drawer to store or takeout food, closing the drawer by force. Depending on the consumer usage conditions, the drawer and handle parts receive repetitive mechanical loads when the consumer opens and closes the drawer.

Figure 4 shows the functional design concept of the drawer and handle system. The stress due to the weight load of the food is concentrated on the handle and support slide rail of the

> *draw load F W* = μ

drawer. Thus, the drawer must be designed to endure these repetitive stresses. The force balance around the drawer and handle system cans be expressed as:

*L x*

β

*B*

λ

( )

⋅*LB* in Equation (9). The failure rate of the final design samples is derived in

( )

β

β

*L*

⋅ ⋅

β

*n h AF*

*r*

λ

missing or improper parameters in the design phase.

**1.4. Refrigerator unit** *LBx* **life and failure rate,** 

the design phase.

λ

Equation (10)

Let *x* =

**system** 

samples using Equation (6).

Repetitive stress can be expressed as the duty effect that carries the on/off cycles and shortens part life [17]. Under accelerated stress conditions, the acceleration factor (AF) can be described as:

$$AF = \left(\frac{S\_\text{\tiny{}}}{S\_\text{\tiny{}}}\right)^{\text{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\langle}}}}}}}}}}}}}}{k}} \left[\frac{E\_a}{T\_0} - \frac{1}{T\_1}\right] \right] = \left(\frac{e\_1}{e\_0}\right)^{\text{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\langle}}}}}}}}}}}}{k}} \left(\frac{E\_a}{k} \left(\frac{1}{T\_0} - \frac{1}{T\_1}\right)\right) \tag{3}$$

And *n* can be determined by multiple testings with different stress levels.

#### **1.3. Parametric ALT with** *BX* **life and sample size**

Traditionally, the characteristic life is defined as:

$$
\eta^{\beta} \equiv \frac{\sum t\_i^{\beta}}{r} \equiv \frac{n \cdot h^{\beta}}{r} \tag{4}
$$

As the reliability of a product (or part) improves, failures of the product become less frequent in laboratory tests. Thus, it becomes more difficult to evaluate the characteristic life using Equation (4). The distribution of failed samples should follow the Poisson distribution for small samples [18]. For a 60% confidence level, the characteristic life can be redefined as

$$
\eta^{\beta} \equiv \frac{1}{r+1} \cdot n \cdot h^{\beta} \tag{5}
$$

In order to introduce the *BX* life in the Weibull distribution, the characteristic life can be modified as

$$L\_B^\beta \equiv \mathbf{x} \cdot \boldsymbol{\eta}^\beta = \frac{\boldsymbol{\chi}}{r+1} \cdot \boldsymbol{n} \cdot \boldsymbol{h}^\beta \tag{6}$$

where *LB* = *BX* life and *x* = 0.01*X*, on the condition that *x* ≤ 0.2.

*BX* is the time by which *X* % of the drawer and handle system installed in a particular population of refrigerators will have failed. In order to assess the *BX* life with about a 60% confidence level, the number of test samples is derived in Eq. (7). That is,

$$m \equiv \frac{1}{\chi} \cdot \left(r + 1\right) \cdot \left(\frac{1}{h^\*}\right)^{\mathcal{Y}} \tag{7}$$

with the condition that the durability target is defined as follows,

$$h^\* = \left(AF \cdot h\right) \Big/ L\_B \ge 1\tag{8}$$

Based on the customer usage conditions, the normal range of operating conditions and cycles of the product (or parts) are determined. Under the worst case, the objective number of cycles and the number of required test cycles can be obtained from Eq. (7). ALT equipment can then be conducted on the basis of load analysis. Using ALT we can find the missing or improper parameters in the design phase.

#### **1.4. Refrigerator unit** *LBx* **life and failure rate,** λ**, with the improved designs**

228 Energy Efficiency – The Innovative Ways for Smart Energy, the Future Towards Modern Utilities

 

 <sup>−</sup>

*k T T*

β η

η

*<sup>S</sup> AF <sup>a</sup>*

0 0 1

And *n* can be determined by multiple testings with different stress levels.

*n*

*E*

*a*

 

 =

*S*

**1.3. Parametric ALT with** *BX* **life and sample size** 

Traditionally, the characteristic life is defined as:

be described as:

modified as

( ) exp ( ) exp *n n a a <sup>f</sup> E E T AS A e*

Repetitive stress can be expressed as the duty effect that carries the on/off cycles and shortens part life [17]. Under accelerated stress conditions, the acceleration factor (AF) can

<sup>1</sup> 1 1 1 1

*it n h r r*

β

 β

> β

β

β

As the reliability of a product (or part) improves, failures of the product become less frequent in laboratory tests. Thus, it becomes more difficult to evaluate the characteristic life using Equation (4). The distribution of failed samples should follow the Poisson distribution for small samples [18]. For a 60% confidence level, the characteristic life can be redefined as

> 1 1 *n h*

In order to introduce the *BX* life in the Weibull distribution, the characteristic life can be

 ≅⋅ = ⋅⋅ η

*BX* is the time by which *X* % of the drawer and handle system installed in a particular population of refrigerators will have failed. In order to assess the *BX* life with about a 60%

> ( ) \* 1 1 *n r* 1 *x h*

≅⋅+⋅

Based on the customer usage conditions, the normal range of operating conditions and cycles of the product (or parts) are determined. Under the worst case, the objective number

*r* β

1 *<sup>B</sup> <sup>x</sup> L x nh r*

ββ

confidence level, the number of test samples is derived in Eq. (7). That is,

with the condition that the durability target is defined as follows,

where *LB* = *BX* life and *x* = 0.01*X*, on the condition that *x* ≤ 0.2.

 ≅ ⋅⋅ +

 

*kT kT*

 

*n*

*E*

 

1

*e e*

 = 

− − = = (2)

 <sup>−</sup>

*k T T*

<sup>⋅</sup> ≡ ≅ (4)

0 0 1

 

(5)

+ (6)

(7)

( ) \* <sup>1</sup> *<sup>B</sup> h AF h L* =⋅ ≥ (8)

(3)

  The parameter design criterion of the newly designed samples can be more than the target life of *BX* = 10 years. From the field data and from a sample under ALT with a corrective action plans, we can obtain the missing or improper parameters of parts and their levels in the design phase.

With the improved design parameters, we can derive the expected *LBx* life of the final design samples using Equation (6).

$$L\_{\rm B}{}^{\beta} \equiv \mathbf{x} \cdot \frac{\mathbf{n} \cdot \left(\hbar \cdot AF\right)^{\beta}}{r+1} \tag{9}$$

Let *x* = λ⋅*LB* in Equation (9). The failure rate of the final design samples is derived in Equation (10)

$$\mathcal{A} \equiv \frac{1}{L\_{\mathcal{B}}} \cdot \left(r + 1\right) \cdot \frac{L\_{\mathcal{B}}^{\mathcal{B}}}{n \cdot \left(h \cdot AF\right)^{\mathcal{B}}} \tag{10}$$
