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-2.5
E-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.2 In 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 105000
E-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.