^{1}

^{2}

As a reliability quantitative specification, parametric accelerated life testing was used to assess the reliability of a newly designed compressor of a commercial refrigerator subjected to repetitive stresses. A generalized life-stress failure model and new sample size equation with a new load concept were derived starting with the basic refrigeration cycle. The sample size equation with the acceleration factor also enabled the parametric accelerated life testing to quickly evaluate the expected lifetime. The design of this testing should help an engineer uncover the design parameters affecting reliability during the design process of the compressor system. Consequently, it should help companies improve product reliability and avoid recalls due to the product failures in the field. A newly designed compressor in a commercial refrigerator was used as a test case.

Reliability describes the ability of a system or module to function under stated conditions for a specified period of time [

as the product reaches the end of its useful life. If a refrigeration system or a major component in the system such as a compressor was to exhibit the failure profile in the bathtub curve with a large number of failures in the early life of its introduction, it would be difficult for the system to be successful in the marketplace. Improving the reliability of a system or component through systematic testing should reduce its failure rate from the traditional failure rate typified by the bathtub curve to the failure rate represented by a flat, straight line with the shape parameter β in

The product reliability function can be quantified from the expected product lifetime L_{B} and failure rate l in

In a practical sense, this proportionality is applicable below about 20 percent of the cumulative failure rate [_{B} and failure rate l by finding the appropriate control parameters affecting reliability and then modifying the design with the results from parametric accelerated life testing (ALT).

In this study we present a new parametric accelerated life testing (ALT) method that can improve the reliability of the design of refrigeration systems. This testing method takes into account the fact that refrigeration system such as compressor with missing or improper design parameters can result in recalls and loss of brand name value. Based on the load analysis of a refrigeration system, we demonstrated the efficacy of our reliability design method. The method uses accelerated life testing and sample size equations, as a novel means of determining proper design parameters.

As can be seen in

_{B} with multi-modules should be determined by module #3 in

Failure data from the field is an important determinant in developing an accelerated life testing plan (ALT). An ALT plan can be developed for a component or module of the system that has a high failure rate or short life.

This plan, if implemented properly, can reduce failure rates and improve reliability. In targeting the reliability of the refrigerator, there are three cases: the new design maintains a similar module, new module, and modified module to the prior design.

For example, consider the list of modules or components in the refrigerator listed in _{Bx} was 4 years from the field data that was collected on the system. Because this was a modified design, the expected failure rate was 0.6%/year and the expected L_{Bx} life was 2.0. Two years was considered an unacceptably short lifetime, so a new target was set to increase the product lifetime to be

No | Reliability | Original design (or market data) | Expected design | Targeted design | |||||
---|---|---|---|---|---|---|---|---|---|

Modules | Yearly failure rate, %/yr | B_{x} life, yr | Yearly failure rate, %/yr | B_{x} life, yr | Yearly failure rate, %/yr | B_{x} life, yr | |||

1 | Cabinet | 0.34 | 5.3 | New | x5 | 1.70 | 1.1 | 0.15 | 12 |

2 | Door | 0.35 | 5.1 | Similar | x1 | 0.35 | 5.1 | 0.15 | 12 |

3 | Internal fixture | 0.25 | 4.8 | Modified | x2 | 0.50 | 2.4 | 0.10 | 12 |

4 | Refrigeration System | 0.30 | 4.0 | Modified | x2 | 0.60 | 2.0 | 0.10 | 12 |

5 | Control | 0.15 | 8.0 | Similar | x1 | 0.15 | 8.0 | 0.1 | 12 |

6 | Others | 0.50 | 12.0 | Similar | x1 | 0.50 | 12.0 | 0.5 | 12 |

Total | Refrigerator | 1.79 | 7.4 | - | - | 3.60 | 3.7 | 1.10 | 12 |

L_{Bx} 12 years with a yearly failure rate over the twelve years to be 1.0. The refrigerator reliability might be determined by summing the failure rates of each module and lifetimes of each module. The refrigerator reliability is targeted to be over a yearly failure rate of 1.1% and B_{x} 12 years.

The mechanisms of failure can be characterized by either loads (or stress) related failure or structural (or materials) related. For example, if there is a void in a shoulder fillet in the structure where the loads are applied, the structure can fracture at that location (see

If one of the modules has a problem due to an improper design, then that module will determine the lifetime of the failed product. Ideally, the system goes from preliminary design to an optimized design in the design process (

A refrigeration system in a refrigerator consists of multiple modules. These modules can be put together as a subassembly and have an input and output for a vapor compressor cycle. As parameters and required input, robust design schematic of refrigeration system employs two experimental arrays: one for the control array (design) and the other for the noise array (loads). Optimizing over the control factors, optimal design data can be reduced to a signal (output)-to-noise (load) ratio. While there are multiple modules in a refrigeration system, ultimately, it is designed to produce cold air (see

If a refrigeration module such as compressor malfunctions, consumers will request that the module be replaced. Refrigeration system engineers often may not have a clear understanding of the way a consumer has used the refrigerator in the field. If the field usage patterns are fully understood, the laboratory could reproduce the usage patterns and failures found in the field. Before launching a product, an engineer should understand the usage patterns that the product will be subjected. If any problems are identified from the expected usage patterns, then the engineer could take corrective actions in the design of the product.

However, it may not be easy to identify all failure modes attributable to the particular design because the failure modes in the mechanical systems come from repetitive stresses which may not be found in its initial testing. Refrigeration system modules need to be robustly designed to withstand a variety of loads. Consequently, the refrigeration system module determines the control factors (or design parameters) to endure the noise factors (or stresses) so the system properly works. Ultimately, this will determine the reliability target of the module or part failure rate l and lifetime L_{B}. Such reliability targeting is known to be conventionally achieved through the Taguchi methods (SDE) and the statistical design of experiment [

For a simple refrigeration system structure, the Taguchi methods should take into account a considerable number of design parameters. In the design process, it may not be possible to consider the whole range of the load conditions that could affect the design. Well-designed parametric ALT methods could help determine the critical parameters affecting reliability of the refrigeration system so it can withstand a wide range of repetitive loads [

To derive the life-stress model and acceleration factor, the time to failure (TF) can be estimated from the McPherson’s derivation [

where S is stress, E_{a} is activation energy, k is Boltzmann’s constant, and T is absolute temperature.

To use Equation (2) for accelerated testing, it needs to be modified and put into a more applicable form. A refri- geration system operates on the basic vapor-compression refrigeration cycle. The compressor receives refrigerant

from the low-side (evaporator) and then compresses and transfers the refrigerant to the high-side (condenser) of the system. The capillary tube controls the flow in a refrigeration system and drops the high pressure of the refrigerant in the condenser to the low pressure in the evaporator. In a refrigeration cycle design, it is necessary to determine both the condensing pressure, P_{c} and evaporating pressure, P_{e} (see

The mass flow rate of refrigerant in a compressor can be modeled as

The mass flow rate of refrigerant in a capillary tube can be modeled as [

By conservation of mass, the mass flow rate can be determined as:

The energy balance in the condenser can be described as

The energy balance in the evaporator can be described as

When nonlinear Equation (5) through (7) are solved, the mass flow rate, _{e}, and condenser temperature, T_{c} can be obtained. Because the saturation pressure, P_{sat}, is a function of temperature, the evaporator pressure, P_{e} (or condenser pressure P_{c}), can be obtained as:

One source of stress in a refrigeration system may come from the pressure difference between suction pressure, P_{suc}, and discharge pressure, P_{dis}.

For a refrigeration system, the time-to-failure, TF, can be modified as

Because the material strength degrades slowly in many mechanical stress/strength interfaces, it may require long testing times until a module failure occurs. The main hurdles to finding wear induced failures and overstressed failures are the testing time and costs incurred from long tests. To resolve these issues, parametric ALT testing under severe conditions would be preferred.

When a refrigeration system structure is subjected to accelerated loads, engineering alloys in the stress-strain curve passes the specification limits (proportional limit), operating limits (elastic limit), yield point, and ultimate stress point until fracture (destruction limit) occurs. In accelerated testing, the appropriate accelerated stress levels (S_{1} or e_{1}) will typically fall outside the specification limits but inside the operating limits of the material (see

From the time-to-failure in Equation 10, the acceleration factor can be defined as the ratio between the needed accelerated stress levels and those found under typical operating conditions. The acceleration factor (AF) can be modified to include the load from Equation (9):

The first term on the right hand side of Equation (11) is the load and the second term is the internal energy. Under severe testing conditions, the refrigeration system subjected to a duty cycle will experience a shortened module lifetime [

The Cumulative Distribution Function (CDF) in the Weibull function can be expressed as:

The Weibull reliability function, R(t), is expressed as:

The characteristic life h_{MLE} from the Maximum Likelihood Estimation (MLE) can be derived as:

If the confidence level is 100(1 − a) and the number of failures, r, is expected to be r ³ 1, then the characteristic life, h_{a}, can be estimated from Equation (14):

Presuming there are no failures, the p-value is a and In(1/a) is mathematically equivalent to the Chi-Squared value,_{a}, would then be represented as:

Equation (15) is established for all cases r ³ 0 and can be redefined as follows:

To evaluate the Weibull reliability function, the characteristic life can be converted into L_{B} life as follows:

After a logarithmic transformation, Equation (18) can be expressed as:

If the estimated characteristic life of p-value a, h_{a}, in Equation (17), is substituted into Equation (19), we obtain the B_{X} life equation:

Because most accelerated lifetime testing typically has a small number of samples, the number of failures would not be as much as that of the sample size.

If Equation (21) is substituted into Equation (20), the B_{X} life equation becomes an inequality and can be written as follows:

The sample size equation with the number of failures can also be modified as follows:

For a 60% confidence level, the first term,

If the acceleration factor, AF, from Equation (11) is added into the planned testing time, Equation (24) can be modified to include AF. When the target reliability-failure rate l and lifetime L_{B} are given, this equation will be used to carry out parametric accelerated life testing:

The suction reed valves of the compressors in domestic refrigerator were cracking and fracturing in the field, as shown in

The compressor designer did not know the reliability of the compressor, either its life, B_{x}, or failure rate, l. To determine these values, a parameter ALT was carried out on the compressor. From Equation (11), assumed the cumulative damage exponent l is 2, the acceleration factor (AF) can be obtained as

System conditions | Worst case, gauge | ALT, gauge | AF | |
---|---|---|---|---|

Pressure, kg/cm^{2} | High side | 13.0 | 30.0 | 5.3 l (=(/k)^{2}) |

Low side | 0.0 | 0.0 | ||

ΔP | 13 | 30k | ||

Temp., ˚C | Dome Temp. | 90 | 120 | 3.92m |

Total AF (=l × m) | - | 20.9 |

The reliability of the new compressor was targeted to be 10 years over B1 that will become the cumulative failure rate 1%. Based on the technical data under the customer usage conditions, the normal number of operating cycles of the suction reed valve for one day was approximately twenty-two; the worst case was ninety-eight. Under the worst case, the expected compressor cycles for ten years would be 357,700 cycles (L_{B}) (see

Assuming the shape parameter β for compressor is common 1.9, the test cycles and test sample numbers calculated in Equation 25 were 39,000 cycles and 20 EA, respectively. As a reliability quantitative specification, parametric ALT was designed to assure a B1 of ten years life with about a 60 percent confidence level of common sense if no unit in it fails during 39,000 cycles.

The ALT equipment was a simplified vapor-compression cycle and fabricated with duty cycles as shown in

The condenser outlet connected the evaporator outlet with the capillary tube. The compressor mounted on the

Item | Number of operations (times) | |||
---|---|---|---|---|

1 day | 10 years | |||

Normal | Worst | Normal | Worst | |

Compressor | 22 | 98 | 80,300 | 357,700 |

rubber pads was connected to the condenser inlet and evaporator outlet. A fan and two 60 Watt lamps maintained the air temperature within the insulated (fiberglass) box. A thermal switch attached on the compressor top controlled the 30 cfm Suntronix axial fan, Model ST1238.

After the system was leak tested, the equipment was evacuated from both high and low sides to 0.01 torr of the static capability of the vacuum pump. It was charged with refrigerant through the valve on the low side of the equipment to an instantaneous pressure of 30 kg/cm^{2}. The test conditions and test limits were set up on the control board. As the test began, the high-side and low-side pressures could be observed on the pressure gauges (or display monitor).

The fracture of the suction reed valves came from its relatively weak structural design. The design included: (1) an overlap with the valve plate, (2) a carbon steel (20C), and (3) a sharp edge on the valve plate (

of the compressor was leaking of refrigerant and locking due to the cracking and fracturing of the suction reed valve.

For the second ALT, the test sample number calculated in Equation (25) was 30 compressors and the mission cycles were 32,000. Three samples failed at 17,000 cycles. However, the calculated life of the newly designed compressor still did not reach the target life of B1 of ten years. From the failure analysis, the root cause of the failed compressor was attributed to a scratch on the crank shaft. The modified design points required heat treating the surface of the crankshaft. To improve compressor reliability, a second ALT was implemented with the modified design.

1^{st} ALT | 2^{nd} ALT | 3rd ALT | |
---|---|---|---|

Initial design | Second design | Final design | |

In 23,000 cycles, fracture of suction reed valve is less than 1. | 8,687 cycles: 1/20 (5.0 %) 8,687 cycles: 19/20 Suspension | 17,000 cycles: 3/30 (10.0 %) 17,000 cycles: 27/30 Suspension | 23,000 cycles: 30/30 OK 29,000 cycles: 30/30 OK |

Suction reed valve structure | |||

Material and specification | SANDVIK 20C 0.178t (Carbon Steel) ®SANDVIK 7C 0.178t (Stainless Steel) | FCD500 + No Heat Treatment ®FCD500 + Heat Treatment |

For the third ALT, the test cycles and test sample numbers calculated in Equation (14) were 23,000 cycles and 60EA. In the third ALT, it was found that the compressor did not crack and fracture until 29,000 cycles of testing.

To improve the reliability design of a refrigeration system, the basic concepts of parametric ALT as a reliability quantitative specification were discussed: 1) setting the overall parametric ALT plan of the refrigerator system; 2) failure mechanics, design and reliability testing; 3) parametric accelerated life testing with an acceleration factor; and 4) derivation of the sample size equation. The failure modes and mechanisms of the mechanical system in the field and parametric ALT may come from the design parameters or design flaws not considered in the original design process. For a sample case study on the design flaws, a refrigerator compressor was studied. Based on the products returned from the field and results of the first ALTs, compressor locking occurred because of the failure of the suction reed valve. The root causes of the failed suction reed valve in the refrigerator were 1) an overlap with the valve plate; 2) a weak material (7C); and 3) the sharp edge of the valve plate.

Based on the second ALT, scratches on the crankshaft were identified as a problem. The additional missing design parameter of the compressor required heat treating on the surface of the crank shaft. After a sequence of ALTs, key design changes were identified and B1 ten year life of the compressor was assured.

Seong-Woo Woo,Dennis L. O’Neal, (2016) Improving the Reliability of a Domestic Refrigerator Compressor Subjected to Repetitive Loading. Engineering,08,99-115. doi: 10.4236/eng.2016.83012

AF Acceleration factor

BX Durability index

C1 Trespan size of the valve plate

C2 Material property of suction reed valve

C3 Ball peening and tumbling process

E_{a} Activation energy, 0.56 eV

F(t) Unreliability or cumulative distribution function (CDF)

F Force, kN

h Testing cycles (or cycles)

h^{*} Non-dimensional testing cycles,

k Boltzmann’s constant, 8.62 × 10^{−5} eV/deg

KCP Key control parameter

KNP Key noise parameter

L_{B} Target B_{X} life and x = 0.01X, on the condition that x £ 0.2

n Number of test samples

N_{1}_{ } Pressure difference, MPa

DP_{1} Pressure difference under accelerated stress conditions, MPa

DP_{0} Pressure difference under normal stress conditions, MPa

P_{c} Condenser pressure, kPa

P_{e} Evaporator pressure, kPa

P_{suc} Compressor suction pressure, kPa

P_{dis} Compressor discharge pressure, kPa

Q Volume flow rate, m^{3}/s

Q_{c} Heat transfer in the condenser, kW

Q_{e} Heat transfer in the evaporator, kW

R Reliability function, target reliability (1 − X*0.01) % life in X = unreliability

r Failed numbers

S Stress

S_{1} Mechanical stress under accelerated stress conditions

S_{0} Mechanical stress under normal conditions

t Use time

T Absolute temperature, K

T_{c} Saturated temperature in the condenser, K

T_{e} Saturated temperature in the evaporator, K

t_{i} Test time for each sample, h

TF Time to failure, h

X Accumulated failure rate, %

x x = 0.01・X, on condition that x £ 0.2

c^{2} Chi-squared distribution

h Characteristic life

l Failure rate, %/yr

l Power index or cumulative damage exponent

a p-value, the estimated probability of rejecting the null hypothesis (H_{0}) when H_{0} is true

b Shape parameter in a Weibull distribution

n Stress dependence,

0 Normal stress conditions

1 Accelerated stress conditions

a Actual

MLE Maximum likelihood estimate