E-2.i APPENDIX E-2. UNCERTAINTY OF ENERGY PARAMETERS EF, RE, UA, AND Pon

TABLE OF CONTENTS E-2.1 ENERGY FACTOR (EF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2.1 E-2.2 RECOVERY EFFICIENCY (RE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2.2 E-2.3 STANDBY HEAT LOSS COEFFICIENT (UA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2.2 E-2.4 RATED INPUT (Pon) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2.3

TABLES Table E-2.1 Uncertainty of Rated Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-2.5E-2.1 APPENDIX E-2. UNCERTAINTY OF ENERGY PARAMETERS EF, RE, UA, AND Pon
E-2.1 ENERGY FACTOR (EF) Estimated uncertainty ranges for the energy factors (EFs) of different design options were provided for the typical tank size of electric (50-gal) and gas (40-gal) water heaters by an LBNL consultant. 1 Ratios were developed to represent the minimum and maximum variations in the EF values of the typical tank sizes. Since each design option is a combination of several single designs, the ratio of a combined design option is assigned to be the largest among the single designs involved. For example, gas-fired water heater Design Option 3 (78% RE and 2" Insulation) is a combination of four single designs, the Analytic Baseline, Heat Traps, 78% RE, and 2" Jacket Insulation. The uncertainty range of Design Option 3 is the largest range of these four single designs. This ensures that the estimated uncertainty ranges include those of every single design. Then, by multiplying the ratios of the different design options by their corresponding EF values, the maximum and minimum EF values of all the standard tank sizes were calculated. The resulting uncertainty ranges for EF are characterized by the minimum, the most-likely, and the maximum, so that the probability distribution can be described by a simple triangular distribution. The uncertainty ranges are also in compliance with the Code of Federal Regulation ruling that specifies that reported EF values must be no higher than the mean value of the sample of the tested units. 2In both electric and gas water heaters, the largest range of the EF is within 5% of the value in our simulation study. For oil water heaters, range of EF is set to 2%. The determination of the ranges of uncertainty for recovery efficiency (RE), standby heat loss coefficient (UA), and rated input (Pon) is made using the Water Heater Analysis Model (WHAM). They are estimated by assessing how big the ranges of these parameters can be to yield the reported ranges of uncertainty on EF independently. To achieve this, the partial derivative of EF to each of the three parameters—RE, UA, and Pon, is computed. The width of the range is reduced slightly to account for the effect on EF when two parameters vary simultaneously. It was found that EF is far less sensitive to Pon than the other two parameters.E-2.2 

## ( ) ( ) RE RE EF UA Pon Pon EF UA max max

max 

### . = ×− + × ××− × × ×− × likely 

### 1 1 2 41094 67 5 1620 41094 2



## ( ) ( ) RE RE EF UA Pon Pon EF UA min min min 

### . = ×− + × ××− × × ×− × likely 

### 1 1 2 41094 67 5 1620 41094 2

UA UA 1 1 2 Pon RE EF 1
67.5 RE Pon 24 41094 1 2 max min 

### = ×− + × −    × ×× −    × likely



### UA UA 1 1 2 Pon RE EF 1
67.5 RE Pon 24 41094 1 2 min max 

### = ×− + × −    × ×× −    × likely E-2.2 RECOVERY EFFICIENCY (RE) The set of equations to solve RE for a given EF is:

E-2.3 STANDBY HEAT LOSS COEFFICIENT (UA) Similarly, the range of uncertainty for UA can be determined by the following set of equations:E-2.3 E-2.4 RATED INPUT (Pon) Since the variation in Pon has an negligible effect on the uncertainty of EF, the determination of the uncertainty is independent of the EF value. Therefore, the minimum and the maximum numbers from the LBNL water heater database were used, as listed below: Table E-2.1 Uncertainty of Rated Input Ponmin Ponlikely Ponmax

Electric Water Heater (30-gal) 4606 15354 18766 Electric Water Heater (other sizes) 12966 15354 18766 Gas Water Heater (30-gal) 26000 30000 40000 Gas Water Heater (40-gal) 28000 40000 60000 Gas Water Heater (50-gal) 32000 50000 75000 Gas Water Heater (75-gal) 55000 75000 75000 Oil Water Heater (32-gal) 84000 90000 105000 Oil Water Heater (50-gal) 105000 105000 105000E-2.4 REFERENCES 1. Minniear, M., Residential Water Heaters: Initial Estimates of Manufacturing Costs and Energy Consumption, November 12, 1997, Minniear Corporation. Panama City, FL. Report No. Task 1 of LBNL Subcontract No. 6466797. 2. Title 10, Code of Federal Regulations, Part 430- Energy Conservation Program for Consumer Products, Section 430.24(e)(1)(ii), January 1 , 1998.