### APPENDIX 8-C. LIFETIME DISTRIBUTIONS

TABLE OF CONTENTS

8-C.1 INTRODUCTION ................................................................................................. 8-C-1

8-C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS ..................... 8-C-1

8-C.3 WATER HEATER LIFETIME DISTRIBUTIONS .............................................. 8-C-2

8-C.4 GAS-FIRED POOL HEATERS LIFETIME DISTRIBUTIONS.......................... 8-C-7

8-C.5 DIRECT HEATING EQUIPMENT LIFETIME DISTRIBUTIONS.................... 8-C-8

LIST OF TABLES

Table 8-C.3.1 Water Heaters............................................................................................. 8-C-2

Table 8-C.4.1 Gas-Fired Pool Heaters.............................................................................. 8-C-7

Table 8-C.5.1 Direct Heating Equipment ......................................................................... 8-C-8

LIST OF FIGURES

Figure 8-C.3.1 Fraction of the Gas-Fired Storage Water Heaters Failing ......................... 8-C-3

Figure 8-C.3.2 Cumulative Lifetime Length of Gas-Fired Storage Water Heaters ........... 8-C-3

Figure 8-C.3.3 Fraction of the Electric Storage Water Heaters Failing............................. 8-C-4

Figure 8-C.3.4 Cumulative Lifetime Length of Electric Storage Water Heaters............... 8-C-4

Figure 8-C.3.5 Fraction of the Oil-Fired Storage Water Heaters Failing........................... 8-C-5

Figure 8-C.3.6 Cumulative Lifetime Length of Oil-Fired Storage Water Heaters ............ 8-C-5

Figure 8-C.3.7 Fraction of the Gas-Fired Instantaneous Water Heaters Failing................ 8-C-6

Figure 8-C.3.8 Cumulative Lifetime Length of Gas-Fired Instantaneous Water

Heaters ....................................................................................................... 8-C-6

Figure 8-C.4.1 Fraction of the Gas-Fired Pool Heater Failing........................................... 8-C-7

Figure 8-C.4.2 Cumulative Lifetime Length of Gas-Fired Pool Heaters........................... 8-C-8

Figure 8-C.5.1 Fraction of the Direct Heating Equipment Failing .................................... 8-C-9

Figure 8-C.5.2 Cumulative Lifetime Length of Direct Heating Equipment ...................... 8-C-9

8-C-i

### APPENDIX 8-C. LIFETIME DISTRIBUTIONS

8-C.1 INTRODUCTION

For each product class, DOE characterized the product lifetime using a Weibull

probability distribution that ranged from the minimum to maximum lifetime estimates described

in chapter 8, Life Cycle Cost and Payback Period Analyses. The Weibull distribution is

recommended for application to lifetime data because it can be shaped to match low, average,

and high values while still allowing some probability of exceeding the high value. 1, 2



### 8-C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS

DOE utilized an approach for calculating the Weibull distribution using product lifetime

data to determine low, average, high value, and percentile of high value. A similar approach is

described in the Crystal Ball Technical Note, which uses the most likely value instead of an

average. 3 

### Available data is used to assign low, average, and high values to a random variable

with unknown distribution parameters.

The Weibull distribution can be defined as:

β ⎛ ⎜ ⎝ x L ⎞ ⎟ ⎠ β

−

− 1 

### ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ β

− 

### x

− L

α 

### f ( x )

= exp

α α

Where:



### L = location



### α = scale



### β = shape

Therefore, the cumulative distribution is:

β ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ x

− L

−

α 

### F ( x )

= 1

− exp

Based on available data, Weibull distribution parameters are specified in the following

manner:

1. The output deviates must be greater than the expert opinion for low value,

2. The average X

avg 

### , must be equal to the average value from the available data,

3. The high value, xb, must correspond to some particular percentile point (e.g., 95 percent,

or 90 percent)

The values for the parameters in the above equations are determined using the approach

outlined in the Crystal Ball Technical Note. 3

8-C-1

### This solution can be checked using Crystal Ball by specifying a Weibull distribution with

the calculated parameters (location, scale, and shape) in an assumption cell and generating a

forecast equal to the assumption. The forecast histogram and statistics verify that the Weibull

distribution matches the desired shape.



### 8-C.3 WATER HEATER LIFETIME DISTRIBUTIONS

Table 8-C.3.1 shows the average, minimum, and maximum lifetime, and maximum

percentile values used to determine the Weibull distribution parameters alpha and beta for water

heaters. The analysis utilized a maximum percentile of 95 to 99 percent.



### Table 8-C.3.1 Water Heaters

Product Class Expert Opinion Values Weibull

Parameters

Minimum

(years) Average

(years) Maximum

(years) Maximum

percentile

(%) Alpha

(scale) Beta

(shape)

Gas-Fired/LPG Storage 6 13 20 95 7.8903 1.9134

Electric Storage 6 13 20 95 7.8903 1.9134

Oil-Fired Storage 6 13 20 95 7.8903 1.9134

Gas-Fired Instantaneous 8 20 30 99 13.4203 3.0897



### Figures 8-C.3.1 to 8-C.3.8 show the Weibull distributions for each water heater product

class.

8-C-20% 2% 4% 6% 8% 10% 12% 14%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years)Probability of Failure (%)



### Figure 8-C.3.1 Fraction of the Gas-Fired Storage Water Heaters

Failing

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years)Probability (%)



### Figure 8-C.3.2 Cumulative Lifetime Length of Gas-Fired Storage

Water Heaters

8-C-30% 2% 4% 6% 8% 10% 12% 14%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years)Probability of Failure (%)



### Figure 8-C.3.3 Fraction of the Electric Storage Water Heaters

Failing

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years)Probability (%)



### Figure 8-C.3.4 Cumulative Lifetime Length of Electric Storage

Water Heaters

8-C-40% 5% 10% 15%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years)Probability of Failure (%)



### Figure 8-C.3.5 Fraction of the Oil-Fired Storage Water Heaters

Failing

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years)Probability (%)



### Figure 8-C.3.6 Cumulative Lifetime Length of Oil-Fired Storage

Water Heaters

8-C-50% 2% 4% 6% 8% 10%

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Age of Equipment (Years)Probability of Failure (%)



### Figure 8-C.3.7 Fraction of the Gas-Fired Instantaneous Water

Heaters Failing

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Age of Equipment (Years)Probability (%)



### Figure 8-C.3.8 Cumulative Lifetime Length of Gas-Fired

Instantaneous Water Heaters

8-C-6

### 8-C.4 GAS-FIRED POOL HEATERS LIFETIME DISTRIBUTIONS

Table 8-C.4.1 shows the average, minimum, and maximum lifetime, and maximum

percentile values used to determine the Weibull distribution parameters alpha and beta for gasfired pool heaters. The analysis utilized a maximum percentile of 99 percent.



### Table 8-C.4.1 Gas-Fired Pool Heaters

Product Class Expert Opinion Values Weibull

Parameters

Minimum

(years) Average

(years) Maximum

(years) Maximum

percentile

(%) Alpha

(scale) Beta

(shape)

Gas-Fired Pool Heaters 3 8 20 99 5.4466 1.3417



### Figure 8-C.4.1 and Figure 8-C.4.2 show the Weibull distributions for pool heaters.

0% 5% 10% 15% 20% 25%

0 2 4 6 8 10 12 14 16 18 20

Age of Equipment (Years)Probability of Failure (%)



### Figure 8-C.4.1 Fraction of the Gas-Fired Pool Heater Failing

8-C-70% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 2 4 6 8 10 12 14 16 18 20

Age of Equipment (Years)Probability (%)



### Figure 8-C.4.2	 Cumulative Lifetime Length of Gas-Fired Pool

Heaters

8-C.5 DIRECT HEATING EQUIPMENT LIFETIME DISTRIBUTIONS

Table 8-C.5.1 shows the average, minimum, maximum lifetime, and maximum percentile

values used to determine the Weibull distribution parameters alpha and beta. The analysis

utilized a maximum percentile of 99 percent.



### Table 8-C.5.1	 Direct Heating Equipment

Product Class Expert Opinion Values Weibull

Parameters

Minimum

(years) Average

(years) Maximum

(years) Maximum

percentile

(%) Alpha

(scale) Beta

(shape)

All DHE Equipment 10 15 20 99 5.6257 2.6548



### Figure 8-C.5.1 and Figure 8-C.5.2 show the Weibull distributions for direct heating

equipment.

8-C-80% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

0 2 4 6 8 10 12 14 16 18 20 22

Age of Equipment (Years)Probability of Failure (%)



### Figure 8-C.5.1 Fraction of the Direct Heating Equipment Failing

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 2 4 6 8 10 12 14 16 18 20 22 24

Age of Equipment (Years)Probability (%)



### Figure 8-C.5.2 Cumulative Lifetime Length of Direct Heating

Equipment

8-C-9

### REFERENCES

1. Barnes, P. R., J. W. Van Dyke, B. W. McConnell, S. M. Cohn, and S. L. Purucker, The

Feasibility of Replacing or Upgrading Utility Distribution Transformers During Routine

Maintenance , 1995. Oak Ridge National Laboratory. Oak Ridge, TN. Report No. Report

No. ORNL-6804/R1. < http://www.ornl.gov/~webworks/cpr/v823/rpt/78562.pdf >

2. Karr, T., Making the Most of Life Test Data . Appliance Magazine , 2003.

< http://www.appliancemagazine.com/print.php?article=197&zone=1&first=1 >

3. Crystal Ball, Technote: Derivation of Weibull Distribution Parameters When the

Minimum, Most Likely Value, and a Percentile are Known, 2007.

< http://www.crystalball.com/support/simulation/cbl_gen_021A.html >

8-C-10