D-2.i

APPENDIX D-2. WATER HEATER ANALYSIS MODEL (WHAM)

TABLE OF CONTENTS

D-2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2-1

D-2.2 ALGORITHM FUNDAMENTALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2-1

D-2.3 DEVELOPMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2-3

D-2.4 CALCULATION OF STANDBY HEAT LOSS COEFFICIENT UA . . . . . . . . . . . D-2-5

D-2.5 SAMPLE CALCULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2-6

TABLES

Table D-2.1 DOE Test Procedure Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . D-2-2

Table D-2.2 WHAM Sample Calculation with Specified Inputs . . . . . . . . . . . . . . . . . . . . D-2-7

Table D-2.3 UA Sample Calculation with Specified Inputs . . . . . . . . . . . . . . . . . . . . . . . D-2-8
D-2.1

APPENDIX D-2. WATER HEATER ANALYSIS MODEL (WHAM)

D-2.1 INTRODUCTION

The Water Heater Analysis Model (WHAM) is a simple energy equation that accounts for

a variety of operating conditions and water heater characteristics when calculating energy

consumption.1

 The WHAM energy calculation is based on assumptions that account for a variety

of field conditions and water heater types. The results of WHAM calculations have been compared

to the results of detailed water heaters simulation programs and show a high level of accuracy in

estimating energy consumption.

D-2.2 ALGORITHM FUNDAMENTALS

WHAM, based on the DOE water heater test procedure, permits the user to minimally

describe both the water heater and the water heater operating conditions.2 The operating conditions

are characterized by four variables: 

• daily draw volume, 

• thermostat setpoint temperature, 

• inlet water temperature, and 

• ambient air temperature.

The water heater is described by:

• rated input (Pon), 

• stand-by heat loss coefficient (UA), and 

• recovery efficiency (RE).

The DOE water heater test procedure includes a method for calculating the Energy Factor

(EF), a measure of efficiency. During the standard test procedure, a 24-hour trial is performed in

which 6 equal draws are made during the first 6 hours of the test for a total of 64.3 gallons (243.4L).

For the remaining 18 hours, the water heater is in stand-by mode and associated energy losses are

measured. The energy consumption during the test is totalled and the mass of water drawn is

summed over the six draws. The temperature of the water entering, leaving, and inside the water

heater tank is recorded. The temperature of the air around the water heater is also noted. Additional

equations appear in the test procedure to correct for any deviation from the standard testing

specifications to guarantee uniformity in testing results in different laboratories. The test is based

on the conditions shown in Table D-2.1.
D-2.2

EF 


M#C

p

#(Ttank  Tinlet)

Qdm

(Eq. D-2.1)

Table D-2.1 DOE Test Procedure Operating Conditions

Daily Draw

gal (liter)

Thermostat Setpoint

Temperature 

(F ((C)

Inlet Water

Temperature

(F ((C)

Ambient Air

Temperature

(F ((C)

64.3 (243.4) 135 (57) 58 (14) 67.5 (49.7)

Stated in general terms, EF is the ratio of the energy output of the water heater to the total

amount of energy delivered to the water heater. More specifically, EF is the added energy content

of the water drawn from the water heater divided by the energy required to heat and maintain the

water at the water heater’s setpoint temperature:

where:

EF 
 energy factor

M 
 mass of water drawn (lbs) (kg)

C specific heat of water, (Btu / lb - (F) (k./kg - (C) p




T water heater thermostat setpoint temperature ((F) ((C) tank 


T the inlet water temperature ((F) ((C) inlet 


Q water heater’s daily energy consumption (Btu) (kWh) dm 


This equation establishes the EF for water heaters. Manufacturers are required by federal law

to determine EF and to label all products with EF. UA and RE are determined in the course of the

test procedure and are used in the test procedure equations to calculate EF.

WHAM is based on a number of assumptions. It assumes that the values for Pon, UA, and

RE do not change significantly when conditions differ from test procedure conditions. Other

assumptions also help to simplify the equation:

• Water temperature in the tank is always at the thermostat setpoint

• Water and air temperatures are constant

• Water density is constant

• Specific heat of water is constant

WHAM only calculates average energy use. Other tools should be used for calculating water

heaters’ response to peak use and for sizing systems.
D-2.3

Qin 
 QrecovQstdby (Eq. D-2.2)

Qrecov 
 Qout

RE (Eq. D-2.3)

D-2.3 DEVELOPMENT

WHAM relies on the following variables, which are defined in the DOE test procedure:

QIn = total water heater energy consumption (Btu/day) (kWh/day)

Qrecov = energy consumed to heat the water drawn from the water heater (Btu/day) (kWh/day)

Qstdby = energy consumed to make up for standby heat loss from the water heater (Btu/day)

(kWh/day)

Qout = heat content of the water drawn from the water heater (Btu/day) (kWh/day)

Hstdby = hours of standby for storage water heaters (hr/day)

RE = recovery efficiency

EF = energy factor

Pon = rated input power (Btu/hr) (kWh/day)

UA = standby heat loss coefficient (Btu/hr - (F) (kWh / (C)

Ttank = tank thermostat setpoint temperature ((F) ((C)

Tin = inlet water temperature ((F) ((C)

Tamb = temperature of ambient air surrounding the water heater ((F) ((C)

vol = volume of water drawn in 24 hours (gal/day) (liter/day)

den = density of water (lb/gal) (kg/L)

Cp = specific heat of water (Btu/lb- (F) (kJ/(kg ∙ (C)

The amount of energy consumed by the water heater (Qin) includes not only the amount of

energy needed to heat the water (Qrecov), but also the amount of energy to replace that which is lost

to the surroundings when no water is being drawn from the tank (Qstdby).

By definition, energy consumed to heat the water drawn from the water heater is equal to the

heat content in the water divided by the RE.

Energy consumption to compensate for standby energy loss from a storage tank water heater

is equal to the difference in the water heater thermostat setpoint and the average ambient air

temperature around the water heater multiplied by the standby heat loss coefficient and the hours of

standby. Note that the standby heat loss coefficient is expressed in terms of energy input to the water

heater. Standby is the time when the water heater is not heating water to recover from a draw. Time
D-2.4

Qstdby 
 UA#(Ttank	Tamb)#Hstdby (Eq. D-2.4)

Hstdby 
 24	 Qrecov

Pon (Eq. D-2.5)

Qout 
 vol#den#C

p

#(Ttank	Tin) (Eq. D-2.6)

EF 
 Qout

Qin

(Eq. D-2.7)

Qin 
 Qout

RE

 UA#(Ttank	Tamb)# 24	 Qout

RE#Pon (Eq. D-2.8)

Qin 


vol#den#C

p

#(Ttank	Tin )

RE ∙ 1  UA#(Ttank	Tamb )Pon

 24 ∙ UA#(Ttank	Tamb

 (Eq. D-2.9)

when the burner is firing or elements are energized to make up for standby heat losses is included

in the hours of standby.

The hours of standby variable is the time the water heater is recovering from a draw. This is

equal to 24 hours minus the ratio of the amount of energy used to heat the water drawn from the

water heater divided by the rated input power of the water heater.

Heat content of the hot water drawn from the water heater is equal to the volume of water

drawn multiplied by the water’s density, specific heat, and the temperature difference between the

water heater setpoint and the inlet water temperature.

The energy factor is equal to the heat content of the water drawn from the heater divided by

the total energy consumption.

Qin can be represented by the parameters from DOE test procedure by substituting for Qrecov

and Qstdby in Eq. D-2.2:

WHAM can now be expressed by expanding Qout, rearranging the terms, and simplifying.

This is the equation to calculate average daily energy consumption.
D-2.5

Qstdby 
 Qin	Qrecov (Eq. D-2.10)

Qstdby 
 Qout ∙ 1

EF

	 1

RE (Eq. D-2.11)

Qstdby 
 UA#(Ttank	Tamb)# 24	 Qout

RE#Pon (Eq. D-2.12)

WHAM is a function of readily available information that can be grouped into three

categories:

• Physical characteristics of water represented by density and specific heat (Cp)

• Water heater operating conditions represented by draw volume and temperatures

• Water heater parameters represented by UA, RE, and Pon

D-2.4 CALCULATION OF STANDBY HEAT LOSS COEFFICIENT UA

UA is a necessary input in the WHAM equation, yet it is not always available. The same

DOE test procedure that established a basis for the WHAM equation can be used for determining

a value for UA.

Since energy consumption to make up for standby losses is equal to the total energy

consumption minus the energy content of water withdrawn from the water heater, it is possible to

rearrange Eq. D-2.2 as follows:

Substituting for Qin and Qrecov from Eqs. D-2.3 and D-2.6, Eq. D-2.10 becomes:

The energy consumption to make up for standby losses can also be defined by combining

Eqs. D-2.3 D-2.4, and D-2.5:

Equating the two terms for Qstdby from Eq. D-2.11 and D-2.12 and solving for UA produces

Eq. D-2.13:
D-2.6

UA 


1

EF

	 1

RE

(Ttank	Tamb )# 24

Qout

	 1

RE#Pon

(Eq. D-2.13)

UA 


1

EF

	 1

RE

67.5# 24

41094

	 1

RE#Pon

(Eq. D-2.14)

Inserting the temperatures and conditions specified in the test procedure, the equation becomes:

This equation allows the user to determine a value for UA, if it is unavailable from other

sources.

D-2.5 SAMPLE CALCULATION

A sample application of the WHAM equation to calculate the energy consumption and the

standby heat loss coefficient (UA) is presented in Table D-2.2 and Table D-2.3. The available data

is organized into three sections representing: 1) the inputs for the energy parameters from DOE test

procedure, 2) the operating conditions parameters, and 3) the physical characteristics of water. The

average daily energy input (energy consumption) and the standby heat loss coefficient are calculated

directly from the input data. Detailed definitions of all the parameters can be found in the DOE Test

Procedure document.2
D-2.7

WHAM Sample Calculation

Inputs:

10 CFR part 430 Testing Procedure Results

UA 3.319 (1.751) Btu/hr-deg F (W/ deg C) Standby Heat Loss Coefficient

RE 0.98 [- ] Recovery Efficiency

Pon 15,354 (4500) Btu/hr (W) Rated Input Power

Operating Conditions

vol 55 (208) gal/day (L/day)

Total Volume of Water Removed 

During the 24 hr Test

Ttank 140 (60) deg F (deg C) Thermostat Set Point

Tin 56 (13) deg F (deg C) Inlet Water Temperature

Tamb 65 (18) deg F (deg C) Ambient Temperature

Physical Characteristiscs of the Water

dens 8.293752 (995.25) lb/gal (kg/cubic meter) Density of the Water

M 456.2 (206.9) lb (kg) Mass of Water Withdrawn During the 24 hr Test

Cp 1.000743 (4.187) Btu/lb-deg F (kJ/(kg - deg C) Specific Heat of Water

===============================================================================

Results:

Average Daily Energy Input

Qin 44468 (13033) Btu/day (W/day) Average Daily Energy Input

where,

Table D-2.2 WHAM Sample Calculation with Specified Inputs
D-2.8

Table D-2.3 UA Sample Calculation with Specified Inputs

Standby Heat Loss Coefficent (UA)

Sample Calculation

Inputs:

10 CFR part 430 Testing Procedure Results

EF 0.700 Energy Factor

RE 0.78 Recovery Efficiency

Pon 40,000 (11.72) Btu/hr (kW) Rated Input Power

Operating Conditions

vol 64.3 (243) gal/day (L/day) Total Volume of Water Removed

During the 24 hr Test

Ttank 135 (57) deg F (deg C) Thermostat Set Point

Tin 58 (14.5) deg F (deg C) Inlet Water Temperature

Qout =M*Cp*(Ttank-Tin) Btu/day (W-h/day) Energy Content of Water

Withdrawn During 24 hr Test

Physical Characteristiscs of the Water at mean of Tin & Tout at test conditions

dens 8.2938 (995.25) lb/gal (kg/cubic meter) Density of the Water

M 456.2 (206.9) lb (kg) Mass of Water Withdrawn

During the 24 hr Test

Cp 1.0007 (4.187) Btu/lb-deg F (kJ/kg - K) Specific Heat of Water

==================================================================

Results:

Standby Heat Loss Coefficient

UA 3.655 (1.928) Btu/hr-deg F (W/deg C)

where,
D-2.9

REFERENCES

1. Lutz, J., C. D. Whitehead, A. Lekov, D. Winiarski, and G. Rosenquist, WHAM: A

Simplified Energy Consumption Equation for Water Heaters. In 1998 ACEEE Summer Study

on Energy Efficiency in Buildings. 1998. Asilomar, CA, August 23-28, 1998: American

Council for an Energy-Efficient Economy. 1: p. 1.171-1.183. 

2. Title 10, Code of Federal Regulations, Part 430- Energy Conservation Program for

Consumer Products, Appendix E to Subpart B- Uniform Test Procedure for Measuring the

Energy Consumption of Water Heaters, January 1, 1998.