Proposed Test Methodology and 

Performance Rating Standard for 

Residential Fuel Cell Systems 

Mark W. Davis 

Michael W. Ellis 

Brian P. Dougherty 

NISTIR 7131
 
 

NISTIR 7131

Proposed Test Methodology and

Performance Rating Standard for 

Residential Fuel Cell Systems 

Mark W. Davis 

Brian P. Dougherty 

Building Environment Division 

Building and Fire Research Laboratory 

Michael W. Ellis 

Department of Mechanical Engineering 

Virginia Polytechnic Institute and State University 

June 2006

U.S. Department of Commerce 

Carlos M. Gutierrez, Secretary 

Technology Administration 

Robert Cresanti, Under Secretary of Commerce for Technology 

National Institute of Standards and Technology 

 William Jeffrey, Director 
 

Executive Summary 

A test method and rating standard is proposed for residential fuel cell systems (RFCSs). The 

proposed approach extends previously developed test procedures for fuel cells by establishing 

specific test methods and calculation procedures that are applicable to residential fuel cell 

systems. These methods and procedures provide the end user with metrics that can be used to 

evaluate the merit of a RFCS in a particular application. 

In the proposed standard, RFCSs are classified into four types: Type I – Grid independent, 

electrical load following; Type II – Grid interconnected, constant power; Type III – Grid 

interconnected, thermal load following; and Type IV – Grid interconnected, water heating. Test 

procedures are described that yield simplified models for the performance of each system based 

on appropriate load characteristics and ambient conditions. 

For Type I systems, the steady fuel use, thermal energy output, and water use are expressed as 

functions of the electrical part-load ratio and the ambient temperature. In addition, the thermal 

energy output, electrical output, fuel use, and water use (for RFCS humidification) associated 

with a hot water simulated use test are evaluated as functions of electrical part-load ratio and 

ambient temperature. Transient electrical load factors are measured and applied to the thermal 

energy output and the fuel consumption to incorporate the system’s performance under transient 

electrical loads. 

For Type II systems, electrical power remains constant and the steady fuel use, thermal energy 

output, and water use are expressed as functions of ambient temperature. In addition, the thermal 

energy output, electrical output, fuel use, and water use (for RFCS humidification) associated 

with a hot water simulated use test are evaluated at rated power and specific values of ambient 

temperature. 

For Type III systems, the system output follows the thermal load and the steady fuel use, 

electrical output, and water use are expressed as functions of thermal part-load ratio and ambient 

temperature. In addition, the electrical output, fuel use, and water use (for RFCS humidification) 

associated with a hot water simulated use test are evaluated at specific values of ambient 

temperature. 

For Type IV systems, the system operates to meet the hot water requirement and the thermal 

energy output, electrical output, fuel use, and water use (for RFCS humidification) associated 

with a hot water simulated use test are evaluated at specific values of ambient temperature. 

For each system type, steady and simulated use models are combined to predict the performance 

of the system in response to typical residential electrical and thermal loads for various 

representative climates. For units that draw ventilation air from the indoors, the performance is 

corrected to account for the energy required to condition outdoor air to replace the ventilation air. 

The annual performance is expressed in terms of the impact on the end user’s net electricity use 

(or generation), fuel use, and water use. 
ii

Table of Contents 

EXECUTIVE SUMMARY...............................................................................i 

TABLE OF CONTENTS.................................................................................ii 

1. PURPOSE .................................................................................................................................1

2. SCOPE.......................................................................................................................................1

2.1 General.............................................................................................................................. 1 

2.2 Interaction with residential energy systems...................................................................... 1 

2.3 Location of installation ..................................................................................................... 1 

3. DEFINITIONS AND NOMENCLATURE..............................................................................2

3.1 Definitions......................................................................................................................... 2 

3.2 Nomenclature.................................................................................................................... 2 

4. CLASSIFICATION ..................................................................................................................8

4.1 Type I: Grid independent, electrical load following RFCS .............................................. 8 

4.2 Type II: Grid interconnected, constant power RFCS........................................................ 8 

4.3 Type III: Grid interconnected, thermal load following RFCS.......................................... 8 

4.4 Type IV: Grid interconnected, RFCS for water heating ................................................... 8 

5. INSTRUMENTS AND DATA ACQUISITION SYSTEMS....................................................8

5.1 Instruments and Methods of Measurement....................................................................... 8 

5.2 Permissible variation in controls and uncertainty limits in measured values ................... 9 

6. APPARATUS ............................................................................................................................9

6.1 General.............................................................................................................................. 9 

6.2 Description of Test Apparatus ........................................................................................ 10 

6.2.1 Overview............................................................................................................... 10

6.2.2 Environmental chamber........................................................................................ 10 

6.2.3 Fuel supply and monitoring system ...................................................................... 10 

6.2.4 Power interface and monitoring system................................................................ 11 

6.2.5 Thermal interface and monitoring system ............................................................ 11 

6.2.6 Power to auxiliary devices.................................................................................... 12 

6.2.7 Exhaust gas monitoring......................................................................................... 12 

7. METHODS OF TEST ............................................................................................................12

7.1 Required tests by fuel cell type....................................................................................... 12 

7.1.1 Simulated-use test ................................................................................................. 12 

7.1.2 Steady-state test .................................................................................................... 13 

7.1.3 Transient electrical power test .............................................................................. 13 

7.2 Environmental conditions for testing.............................................................................. 13 

8. TEST PROCEDURES ............................................................................................................14

8.1 Hot water simulated-use test........................................................................................... 14 

8.1.1 Operating conditions............................................................................................. 14 

8.1.2 Initialization. ......................................................................................................... 14 

8.1.3 Water heating simulated-use test procedure ......................................................... 14 

8.1.4 Test results ............................................................................................................ 15 

8.2 Steady-state test .............................................................................................................. 15 

8.2.1 Operating conditions............................................................................................. 15 
iii

8.2.2 Initialization. ......................................................................................................... 15 

8.2.3 Test results ............................................................................................................ 16 

8.3 Transient electrical power test ........................................................................................ 16 

8.3.1 Operating conditions............................................................................................. 16 

8.3.2 Initialization and test procedure............................................................................ 16 

8.3.3 Test results ............................................................................................................ 16 

8.4 Data acquisition .............................................................................................................. 17 

8.5 Safety .............................................................................................................................. 18 

9. DATA ANALYSIS...................................................................................................................19

9.1 General description ......................................................................................................... 19 

9.2 Calculation of Mass and Energy Flow Rates.................................................................. 19 

9.2.1 Fuel use rate .......................................................................................................... 19

9.2.2 Net electrical power .............................................................................................. 19 

9.2.3 Steady-state thermal output rate............................................................................ 20 

9.2.4 Simulated-use thermal output rate ........................................................................ 20 

9.2.5 Thermal loss rate................................................................................................... 20 

9.2.6 Ventilation airflow rate......................................................................................... 21 

9.2.7 Water use .............................................................................................................. 22

9.3 Transient Electrical Load Factors ................................................................................... 23 

10. CALCULATION OF SYSTEM PERFORMANCE RATINGS ............................................25

10.1 General Description ........................................................................................................ 25 

10.2 Rated and annual performance for grid independent, electrical load following Type I 

RFCS....................................................................................................................................... 26 

10.2.1 Rated performance for outdoor Type I RFCS....................................................... 26 

10.2.2 Rated performance for indoor Type I RFCS......................................................... 26 

10.2.3 Part-load performance evaluation for a Type I RFCS .......................................... 27 

10.2.4 Analysis of Simulated Use Data for Type I RFCS ............................................... 31 

10.2.5 Annual Performance Rating Procedure for Type I RFCS .................................... 33 

10.3 Rated and annual performance for grid interconnected, constant power Type II RFCS 37 

10.3.1 Rated performance for outdoor Type II RFCS ..................................................... 37 

10.3.2 Rated performance for indoor Type II RFCS ....................................................... 37 

10.3.3 Fuel use, thermal output, ventilation airflow, and water use rates as functions of 

ambient temperature for Type II RFCS ................................................................................ 37 

10.3.4 Analysis of Simulated Use Data for Type II RFCS.............................................. 39 

10.3.5 Annual Performance Rating Procedure ................................................................ 40 

10.4 Rated and annual performance for grid interconnected, thermal load following Type III 

RFCS....................................................................................................................................... 44 

10.4.1 Rated performance for outdoor Type III RFCS.................................................... 44 

10.4.2 Rated performance for Type III RFCS ................................................................. 44 

10.4.3 Fuel use, thermal output, ventilation airflow, and water use rates as functions of 

thermal output and ambient temperature for Type III RFCS................................................ 44 

10.4.4 Analysis of Simulated Use Data for Type III RFCS............................................. 48 

10.4.5 Annual Performance Rating Procedure for Type III RFCS.................................. 51 

10.5 Rated and annual performance for grid interconnected, water heating only Type IV 

RFCS....................................................................................................................................... 57 

10.5.1 Rated performance for outdoor Type IV RFCS.................................................... 57 
iv

10.5.2 Rated performance for Type IV RFCS ................................................................. 57 

10.5.3 Analysis of Simulated Use Data for Type IV RFCS ............................................ 57 

10.5.4 Annual Performance Rating Procedure for Type IV RFCS.................................. 60 

11. REFERENCES .......................................................................................................................63

APPENDICES ............................................................................................................................A-1

A. RESIDENTIAL LOADS AND CLIMATE DATA FOR DEPARTMENT OF ENERGY 

(DOE) CLIMATE REGIONS.....................................................................................................A-1

B. EXAMPLE CALCULATION ...............................................................................................B-1

B.1 General Description ...................................................................................................... B-1 

B.2 Nominal performance of a thermal load following RFCS (Type III)........................... B-2 

B.3 Part-Load Functions...................................................................................................... B-2 

B.4 Simulated Use Data....................................................................................................... B-4 

B.5 Energy Ratings.............................................................................................................. B-5 
1 

1. Purpose 

Fuel cells are emerging as one of the most promising technologies for meeting the nation’s 

energy needs. Fuel cells can provide both electricity and thermal energy and can offer the 

benefits of combined heat and power systems at sizes compatible with residential applications. 

Residential fuel cell systems (RFCSs) are currently under development by a number of 

manufacturers and in a variety of system configurations. As these systems approach 

commercialization, it is essential that codes, standards, and rating methodologies be developed to 

support the safe adoption of the technology and to help prospective owners evaluate the benefits 

of a relatively complex system. 

Currently, the American Society of Mechanical Engineers (ASME), American National 

Standards Institute (ANSI), National Fire Protection Association (NFPA), and other standards 

organizations are developing standards applicable to a wide range of fuel cells and applications. 

The ANSI [1] and NFPA [2] standards address the safe operation, construction, installation, and 

acceptable performance of fuel cell units in general. The ASME standard [3] defines a test 

procedure for determining the steady performance of a fuel cell system at a single operating 

point. 

The procedure proposed here provides a method of test and a rating methodology to capture the 

performance of a fuel cell system operating in a residential application [4]. The procedure 

determines the performance of the RFCS in a series of steady and simulated-use tests. Results 

from the tests are used to predict the useful electrical and thermal energy supplied by the fuel cell 

system as well as the fuel used by the system as it interacts with the energy needs of a typical 

residence in a specified geographic location. Information provided by the rating procedure will 

help the prospective owner assess the economic value of the fuel cell system for a specific 

application. 

2. Scope 

2.1 General 

The test methodology and performance rating standard applies to packaged residential fuel cell 

systems fueled by natural gas, propane, or methanol. Packaged systems include fuel reformers, 

air compressors, humidifiers, power conditioners, heat rejection devices, heat exchangers for 

heat delivery systems, condensers, and all other necessary devices and controls to operate as a 

stand-alone system. 

2.2 Interaction with residential energy systems 

The standard applies to systems that receive natural gas and supply alternating current electricity 

to the residence and/or the utility grid. If thermal energy is provided to the residence, it is 

supplied by a circulating thermal energy delivery circuit. 

2.3 Location of installation 

The standard applies to systems intended for both indoor and outdoor installation. 
2

3. Definitions and Nomenclature 

3.1 Definitions 

Circulating fluid – mixture of anti-freeze (specified here to be propylene glycol) and water that 

circulates between the RFCS and external heat exchangers and that is used to transfer thermal 

energy from the RFCS to an external load. 

Constant power RFCS – a residential fuel cell system that supplies power at a constant rate 

regardless of the electrical or thermal loads of the residence. 

Electrical load following RFCS – a residential fuel cell system that is designed to vary its 

electrical output in response to the changing electrical energy needs of the residence. 

Grid independent RFCS – a residential fuel cell system that incorporates power management 

systems and/or electrical storage systems to meet the transient residential electrical loads so that 

interconnection with the utility grid is not necessary. 

Grid interconnected RFCS – a residential fuel cell system that must be connected to the utility 

grid. Grid interconnected systems must meet all applicable codes for interconnection with the 

utility grid. 

Residential fuel cell system (RFCS) – a packaged system that includes a fuel reformer, air 

compressor, humidifiers, power conditioner, heat rejection devices, heat exchangers for heat 

delivery systems, and all other necessary devices and controls to operate as a stand-alone system. 

Residential fuel cell systems receive a hydrocarbon fuel and supply alternating current electricity 

to a residence. In addition, thermal energy may be supplied to the residence and excess 

electricity may be supplied the utility grid. 

Domestic hot water simulated-use test – a test in which water is withdrawn in specified 

quantities and at specified times from the preheat tank connected to the RFCS. The draws are 

scheduled in accordance with a standard procedure and designed to simulate a residential hot 

water use pattern. 

Steady-state test – a test in which the operating conditions including ambient conditions, fuel 

supply, and energy loads remain constant for the duration of the test. 

Thermal load following RFCS – a residential fuel cell system that is designed to vary its thermal 

output in response to the changing thermal energy needs of the residence. 

Real power - time average of the instantaneous product of voltage and current. 

3.2 Nomenclature 

AE annual net electrical energy supplied by the RFCS to the residence, kW⋅h (Btu) 

AF annual natural gas usage by the RFCS, kW⋅h (Btu) 

AG annual net electrical energy supplied by the RFCS to the utility grid, kW⋅h 

AQwh annual thermal energy supplied by the RFCS for water heating, kW⋅h (Btu) 

AQsh annual thermal energy supplied by the RFCS for space heating, kW⋅h (Btu) 

AW annual domestic water usage by the RFCS, L (gal) 

ch the specific heat of the heat transfer fluid used to transfer thermal energy from the 

RFCS to an external load, evaluated at the average of the RFCS inlet and outlet 

temperatures, kJ/kg⋅oC (Btu/lbm⋅oF) 
3

cc the specific heat of water evaluated at the average of the cold water inlet and 

preheat tank outlet temperatures, kJ/kg⋅oC (Btu/lbm⋅oF) 

R E& power at rating conditions (for an outdoor unit), kW 

R* E& power at rating conditions (for an indoor unit), kW 

Efc & net electrical power supplied by the RFCS after deducting power to external 

auxiliary devices, kW 

* Efc & net electrical power supplied by the RFCS after deducting power to external 

auxiliary devices (for indoor units), kW 

Ein & electrical power supplied to RFCS, kW 

Eout & electrical power supplied by the RFCS, kW 

Eaux & electrical power supplied to auxiliary devices not powered by the RFCS, kW 

Ei net electrical energy supplied to the residence during hour i, kW⋅h 

E1i first approximation to the net electrical energy supplied to the residence during 

hour i, kW⋅h 

E electrical energy re L,i quirement of the residence during hour i, kW⋅h 

Ewh,i electrical energy provided by the RFCS as a result of meeting the water heating 

load during hour i at ambient temperature Ti, kW⋅h 

* Ewh,i electrical energy provided by the RFCS as a result of meeting the water heating 

load during hour i at ambient temperature TI, kW⋅h 

EwhH,i electrical energy provided by the RFCS as a result of meeting the water heating 

load during hour i at ambient temperature TH, kW⋅h 

EwhL,i electrical energy provided by the RFCS as a result of meeting the water heating 

load during hour i at ambient temperature TL, kW⋅h 

F& rate at which fuel energy is used by the RFCS, kW (Btu/h) 

R F& rate at which fuel energy is used by the RFCS at rated conditions (for an outdoor

unit), kW (Btu/h) 

R* F& rate at which fuel energy is used by the RFCS at rated conditions (for an indoor

unit), kW (Btu/h) 

Fi fuel used during hour i, kW⋅h (Btu) 

Fss

& the average rate of fuel use by the RFCS during the steady state test, kW (Btu/h) 

Fwh,i fuel used by the RFCS as a result of meeting the water heating load during hour i 

at ambient temperature Ti (for an outdoor unit), kW⋅h (Btu) 

* Fwh,i fuel used by the RFCS as a result of meeting the water heating load during hour i 

at ambient temperature TI (for an indoor unit), kW⋅h (Btu) 

FwhH,i the fuel used by the RFCS as a result of meeting the water heating load during 

hour i at ambient temperature TH, kW⋅h (Btu) 
4

FwhL,i fuel used by the RFCS as a result of meeting the water heating load during hour i 

at ambient temperature TL, kW⋅h (Btu) 

Gi electrical energy supplied to the utility grid during hour i, kW⋅h 

0 HHV higher heating value of fuel f gas at standard conditions (T0

; P0), kJ/L (Btu/ft3) 

i index for hours 

Ktf transient electrical load correction factor for the fuel use 

Ktq transient electrical load correction factor for the thermal output 

Mfuel the molar mass of the fuel 

R* ma & mass flow rate of ventilation air at rating conditions (for an indoor unit that uses 

indoor air for ventilation), kg/h (lbm/h) 

on ma & mass flow rate of air to the fuel cell while fuel is reacting, kg/h (lbm/h) 

off ma & mass flow rate of air to the fuel cell while the fuel is not reacting (e.g., during preand post-purge periods), kg/h (lbm/h) 

* ma,i total ventilation air flow quantity (associated with electrical output, space heating, 

and/or water heating) for hour i , kg (lbm) 

pa absolute pressure of air surrounding the RFCS, kPa (psia) 

pf absolute pressure of fuel gas at the flow rate monitoring point, kPa (psia) 

ps(T) the saturation pressure of water at T, kPa (psia) 

pv pressure of the exhaust gas at the flow rate monitoring point, kPa-absolute (psia) 

p0

 is standard pressure, 101.3 kPa (14.7 psia) 

Qwh & the useful thermal energy provided for water heating by the RFCS in combination 

with the specified preheat tank during the simulated use test (i.e. the useful 

thermal energy delivered by the preheat tank to the water heater tank), kW (Btu/h) 

Qfc,i the thermal energy provided by the fuel cell for water heating and space heating 

during hour i, kW⋅h (Btu) 

Qfc & thermal output rate from the RFCS, kW (Btu/h) 

Qfc,ss & the average thermal output by the RFCS during the steady state test, kW (Btu/h) 

Qloss,i the thermal energy lost in the circulating fluid between the RFCS and the preheat 

tank during hour i, kW⋅h (Btu) 

Qwh,i the thermal energy provided for water heating during hour i, kW⋅h (Btu) 

R Q& thermal output rate at rated conditions (for an outdoor unit), kW (Btu/h) 

R* Q& thermal output rate at rated conditions (for an indoor unit), kW (Btu/h) 

raf mass basis air-to-fuel ratio 

re average electrical part-load ratio – the ratio of the electrical output during an hour 

to the rated electrical output for an hour 

r1e first approximation to the average electrical part-load ratio 
5

rt average thermal part-load ratio - the ratio of the thermal output during an hour to 

the rated thermal output for an hour 

RHa relative humidity of the ambient air, % 

RHv relative humidity of the exhaust air from the RFCS, % 

Ta temperature of air surrounding the RFCS, oC (oF) 

Tci temperature of cold water entering the preheat tank, oC (oF) 

Tf absolute temperature of fuel gas at the flow rate monitoring point, K (R) 

The temperature of circulating fluid at the exit from the RFCS, oC (oF) 

Thi temperature of circulating fluid at the inlet to the RFCS, oC (oF) 

Tp temperature of water in preheat tank, oC (oF) 

Tpe temperature of water leaving the preheat tank, oC (oF) 

Tv absolute temperature of the exhaust gas at the point where the flow rate is 

measured, K (R) 

T0

 standard temperature, 298 K (537 R) 

τ duration of the steady state test, hours 

Vc

& volumetric flow of cold water into the preheat tank, L/s (ft3

/h.) 

Vf

& volumetric flow rate of fuel gas, L/h (ft3

/h) 

Vh

& volumetric flow rate of the heat transfer fluid through the RFCS, L/s (ft3

/h) 

Vv

& the volumetric flow rate of the exhaust gas, L/h (ft3

/h) 

Vw

& flow rate of domestic water to the fuel cell (i.e. to supply the humidification 

system), L/h (gal/h) 

R Vw

& water use rate at rated conditions (for an outdoor unit), L/h (gal/h) 

R* Vw

& water use rate at rated conditions (for an indoor unit), L/h (gal/h) 

V total volume of water used w,i (associated with electrical output, space heating, 

and/or water heating for an outdoor unit) for hour i , L (gal) 

* Vw,i total volume of water used (associated with electrical output, space heating, 

and/or water heating for an indoor unit) for hour i , L (gal) 

Vw,wh,i water used by the RFCS as a result of meeting the water heating load during hour 

i at ambient temperature Ti (for an outdoor unit), L (gal) 

* Vw,wh,i water used by the RFCS as a result of meeting the water heating load during hour 

i at ambient temperature TI (for an indoor unit), L (gal) 

Vw,whH,i water used by the RFCS as a result of meeting the water heating load during hour 

i at ambient temperature TH (for an outdoor unit), L (gal) 

Vw,whL,i water used by the RFCS as a result of meeting the water heating load during hour 

i at ambient temperature TL (for an outdoor unit), L (gal) 
6

Xq part-load thermal output factor for electrical load following applications that 

indicates the fraction of the rated thermal output that occurs at electrical part-load 

ratio, re, and ambient temperature Ta (for an outdoor unit) 

* Xq part-load thermal output factor for electrical load following applications that 

indicates the fraction of the rated thermal output that occurs at electrical part-load 

ratio, re, and ambient temperature TI (for an indoor unit) 

Xf part-load fuel use factor for electrical load following applications that indicates 

the fraction of the rated fuel use rate that occurs at electrical part-load ratio, re, 

and ambient temperature Ta (for an outdoor unit) 

* X part-load fuel use factor for electrical f load following applications that indicates 

the fraction of the rated fuel use rate that occurs at electrical part-load ratio, re, 

and ambient temperature TI (for an indoor unit) 

* X part-load ventilation airflow factor for el v ectrical load following applications that 

indicates the fraction of the rated ventilation airflow that occurs at electrical partload ratio, re, and ambient temperature TI (for an indoor unit that uses indoor air 

for ventilation) 

Xw part-load water use factor for electrical load following applications that indicates 

the fraction of the rated water use that occurs at electrical part-load ratio, re, and 

ambient temperature Ta (for an outdoor unit) 

* X part-load water use factor for electrical w load following applications that indicates 

the fraction of the rated water use that occurs at electrical part-load ratio, re, and 

ambient temperature TI (for an indoor unit) 

dry Y volume fraction of CO CO2 2 in dry exhaust gas 

0 Y volume fraction of CO CO2 2 in ambient air 

Ze part-load power output factor for thermal load following applications that 

indicates the fraction of the rated power output that occurs at thermal part-load 

ratio, rt, and ambient temperature Ta (for an outdoor unit) 

* Z part-load power output factor for ther e mal load following applications that 

indicates the fraction of the rated power output that occurs at thermal part-load 

ratio, rt, and ambient temperature TI (for an indoor unit) 

Zf part-load fuel use factor for thermal load following applications that indicates the 

fraction of the rated fuel use rate that occurs at thermal part-load ratio, rt, and 

ambient temperature Ta (for an outdoor unit) 

* Z part-load fuel use factor for thermal load f following applications that indicates the 

fraction of the rated fuel use rate that occurs at thermal part-load ratio, rt, and 

ambient temperature TI (for an indoor unit) 

* Zv part-load ventilation airflow factor for thermal load following applications that 

indicates the fraction of the rated ventilation airflow that occurs at thermal partload ratio, rt, and ambient temperature TI (for an indoor unit that uses indoor air 

for ventilation) 
7

Zw part-load water use factor for thermal load following applications that indicates 

the fraction of the rated water use that occurs at thermal part-load ratio, rt, and 

ambient temperature Ta (for an outdoor unit) 

* Z part-load water use factor for thermal lo w ad following applications that indicates 

the fraction of the rated water use that occurs at thermal part-load ratio, rt, and 

ambient temperature TI (for an indoor unit) 

0 ρa density of air at standard conditions (T0

; P0), kg/L (lbm/ft3) 

ρc density of water at Tci, kg/L (lbm/ft3

) 

ρh density of the heat transfer fluid evaluated at the temperature of the fluid at the 

flow measurement point, kg/L (lbm/ft3) 

0 ρf density of the fuel at standard conditions (T0

; P0), kg/L (lbm/ft3) 

ωa humidity ratio of the ambient air, kg-water/kg-dry air (lbm-water/lbm-dry air) 
8

4. Classification 

Residential fuel cell systems may be classified into four types according to the interaction of the 

system with the electrical grid and with the thermal load for the residence. 

4.1 Type I: Grid independent, electrical load following RFCS 

Residential fuel cell systems for grid independent applications strive to meet the electrical loads 

of the residence at all times using either power supplied directly by the fuel cell stack or using 

supplemental power supplied by storage sub-systems (e.g., batteries or capacitors) that are part of 

the fuel cell system. Thermal energy produced during fuel cell system operation can be supplied 

to the residence for space heating and/or domestic water heating1

. 

4.2 Type II: Grid interconnected, constant power RFCS 

Residential fuel cell systems for grid interconnected applications can use power from the utility 

grid to supplement fuel cell power during periods of high electrical load by the residence and can 

supply power to the grid during periods of low electrical load. In addition, the fast response of 

the electrical grid can be used to meet transient load conditions. Thus grid connected fuel cell 

combined heat and power (CHP) systems can operate to produce constant electrical power. As 

with the grid independent system, thermal energy produced during fuel cell system operation can 

be supplied to the residence for space heating and/or water heating. 

4.3 Type III: Grid interconnected, thermal load following RFCS 

Residential fuel cell systems for grid interconnected thermal load following applications control 

the fuel cell operation in an attempt to satisfy the thermal load of the residence including the 

space heating load and/or the domestic water heating load. The coincident power production 

contributes to meeting the electrical load of the residence. Power from the grid is used to 

supplement the fuel cell power as required to meet the electrical load. If the coincident power 

exceeds the electrical load of the residence, the excess is transferred to the grid. 

4.4 Type IV: Grid interconnected, RFCS for water heating 

Residential fuel cell systems for grid interconnected water heating applications control the fuel 

cell operation to satisfy the water heating load. As with thermal load following applications, the 

coincident power production contributes to meeting the electrical load of the residence. Power 

from the grid is used to supplement the fuel cell power as required to meet the electrical load. If 

the coincident power exceeds the electrical load of the residence, the excess is transferred to the 

grid. 

5. Instruments and Data Acquisition Systems 

5.1 Instruments and Methods of Measurement 

Instruments and methods of measurement shall be in accordance with the ASME Performance 

Test Code for Fuel Cell Power Systems [3]. 

 

1

 Although beyond the scope of this work, the thermal energy may also be used to power cooling devices such as 

absorption air conditioning. 
9

5.2 Permissible variation in controls and uncertainty limits in measured values 

Permissible variations for specific control parameters are specified in Tables 8.1 and 8.2. 

An uncertainty analysis shall be conducted in accordance with the principles outlined in ASME 

PTC-50-2002, Fuel Cell Power Systems Performance [3], Appendix I and in ASME PTC 19.1-

1988, Test Uncertainty [5]. Trade-offs may be made between individual instrument uncertainties 

as long as the overall measurement uncertainty in each value is within 2 %. 

6. Apparatus 

6.1 General 

Fuel cell systems are complex systems that are comprised of multiple sub-systems and that use 

and yield mass and energy streams in a variety of forms. Figure 6.1 illustrates a generic fuel cell 

system and depicts the interaction of the system with the surroundings. For the purpose of this 

test procedure and rating methodology, the mass and energy streams of interest are: 

− Electrical energy provided by the fuel cell system to the residence 

− Electrical energy supplied by the residence to the fuel cell system 

− Electrical energy supplied to external auxiliary devices (e.g., circulating water pumps) 

− Thermal energy provided by the fuel cell system to the residence 

− Fuel used by the fuel cell system 

− Air extracted from the residence by the RFCS 

− Water consumed by the fuel cell system 

Figure 6.1 Residential fuel cell system and interactions with surroundings. 

Water

Mgmt 

Fuel 

Processor

Air

Mgmt

Fuel in 

Water 

Air Heat 

FUEL CELL 

STACK Electricity Out 

Exh 

Heat/Exhaust Out

Fuel 

Power

Conditioner 

Water heating Thermal

Mgmt 

Space heating

Air 

Residential Fuel Cell 

System Boundary 

Electricity Into 

Fuel Cell System

Electricity Into Auxiliary 

Devices for Heat Delivery 

Water 
10

The test apparatus is designed to measure the interactions of the fuel cell system with the 

surroundings so that the fuel use and the net contribution of electrical and thermal energy to the 

residence can be determined. 

6.2 Description of Test Apparatus 

6.2.1 Overview 

As illustrated in Figures 6.2 and 6.3, the test apparatus consists of an environmental 

chamber; a fuel supply and monitoring system; a power interface and monitoring system; a 

thermal energy interface and monitoring system; and a system for monitoring exhaust gas 

CO2 content. The thermal energy supplied by the fuel cell is either transferred to a 

residential domestic water heater (as depicted in Figure 6.2 for the simulated-use test) or 

transferred to a circulating chilled water loop controlled to maintain a constant inlet water 

temperature to the fuel cell system (as depicted in Figure 6.3 for the steady-state test). 

6.2.2 Environmental chamber 

The environmental chamber must be capable of maintaining the temperature and relative 

humidity within the chamber at the specified test conditions while compensating for the 

thermal loads imposed by operation of the fuel cell system. 

6.2.3 Fuel supply and monitoring system 

Natural gas is supplied to the fuel cell system, and the temperature, pressure, and flow rate 

of the natural gas are monitored. In addition, the higher heating value of the fuel is 

determined by a calorimeter at the beginning and end of each test to confirm that the 

heating value is within acceptable limits. 

Figure 6.2. Test apparatus for hot water simulated-use test. 

Residential Fuel 

Cell System

Thermal 

Loop 

In

Out

Bi-directional 

Power Monitor

Tp 

pf Tf 

YCO2

Tv 

Air Intake 

Power In/Out

Exhaust

Tci 

Calorimeter 

Fuel 

Supply 

Ta pa RHa 

Fuel Inlet 

Environmental 

Chamber 

Water 

Heater 

Electrical grid or 

electronic load 

Preheat 

Tank 

Ein &

0 Hf

Eout &

Eaux &

Vc

&

Vf

&

Tpe 

Vv

&

pv

Thi 

The 

RHv
11

Figure 6.3. Test apparatus for steady-state test. 

6.2.4 Power interface and monitoring system 

A monitoring system will be implemented to measure the real power provided by the fuel 

cell to an electronic load or to the utility grid. Any power supplied from the grid to the fuel 

cell for start-up or for operation of auxiliary devices will also be monitored. 

6.2.5 Thermal interface and monitoring system 

A thermal interface consisting of a preheat tank, a water heater, and a heat exchanger are 

connected to the fuel cell system. Any external pumps required for system operation are 

supplied by the manufacturer and installed in accordance with the manufacturer’s 

recommendations to circulate heat transfer fluid through the preheat tank and/or the heat 

exchanger. The characteristics of the thermal interface are specified in Table 6.1. 

Table 6.1. Thermal interface characteristics and maximum permissible variation 

Parameter Value and allowable variation 

Preheat tank volume 303 L (80 gal) 

Preheat tank recovery (32 oC (90 oF) rise) 200 L/h (54 gal/h) 

Preheat tank first hour delivery (32 oC (90 oF) rise) 475 L (125 gal) 

Circulating fluid composition Water and propylene glycol (PpG); 

35 % PpG by volume ±3 % PpG 

Circulating fluid temperature and flow rate See Table 8.1 

Residential Fuel 

Cell System

Thermal 

Loop 

In

Out

The 

Thi 

pf Tf 

Air Intake 

Power In/Out

Calorimeter 

Fuel 

Supply 

Ta pa RHa 

Fuel Inlet 

Environmental 

Chamber 

Electricity In 

Chilled 

water 

Heat 

Exchanger 

Electrical grid or 

electronic load 

Bi-directional 

Power Monitor

0 Hf

Ein & Eout &

Eaux &

Vh

&

Vf

&

YCO2

Tv 

Exhaust

Vv

&

pv

RHv
12

In the simulated-use test, illustrated in Figure 6.2, circulating fluid from the fuel cell system 

is circulated through a double-wall heat exchanger located in the preheat tank. Cold water 

is supplied from the domestic cold water supply line to the preheat tank. Water from the 

preheat tank flows to the hot water tank to replace water extracted during the specified 

draws. Facilities are provided to maintain the cold water inlet temperature to the preheat 

tank within the temperature range specified in Table 8.1. The mean tank temperature for 

the preheat tank and the hot water tank are monitored by temperature probes. Controls are 

installed and operated as recommended by the manufacturer to control the preheat tank 

temperature. In addition to these controls, a means shall be provided by the manufacturer 

to stop the transfer of heat from the RFCS to the preheat tank if the preheat tank exceeds 

the high temperature limit. The temperature and flow rate of water entering the preheat 

tank and the temperature of water leaving the preheat tank are monitored to determine the 

thermal energy supplied by the fuel cell system to the preheat tank. 

In the steady-state test, illustrated in Figure 6.3, circulating fluid from the fuel cell system 

will be circulated through a heat exchanger. The flow of chilled water supplied to the other 

side of the heat exchanger is controlled to maintain the temperature of the water supplied to 

the fuel cell within the specified range. The temperature and flow rate of circulating fluid 

entering the heat exchanger and the temperature of circulating fluid leaving the heat 

exchanger are monitored to determine the thermal energy supplied by the fuel cell system. 

6.2.6 Power to auxiliary devices 

Any power provided to the fuel cell system or used to operate external pumps or auxiliary 

devices shall be measured using real power meters, recorded, and subtracted from the fuel 

cell system output power to determine the net power. 

6.2.7 Exhaust gas monitoring 

The temperature of the exhaust gas is monitored, and instrumentation is provided to 

withdraw a sample of the exhaust gas, dry the sample, and measure the CO2 content of the 

sample. The sample will be withdrawn downstream of any mixing dampers in the exhaust 

system so that the sample reflects the diluting effect of any air withdrawn from the 

environmental chamber by the fuel cell system operation. The CO2 content will be used in 

conjunction with the fuel flow rate to determine the inlet air flow rate. 

7. Methods of Test 

7.1 Required tests by fuel cell type 

Each fuel cell type requires a specific set of tests to evaluate its performance. The tests consist 

of two types: 

7.1.1 Simulated-use test 

The simulated use test is a 24 h transient test that represents the energy used for water 

heating. Water is withdrawn from the water heater (and thus the preheat tank) according to 

a schedule that is similar to the Department of Energy Test Procedure for Water Heaters5

. 

The thermal energy supplied by the RFCS/preheat tank system to the incoming water is 

monitored. The simulated-use test can be conducted while the fuel cell system is supplying 

electricity at a specific power level or can be conducted in a thermal load following mode 
13

in which the fuel cell system operates only as needed to heat the water. The simulated-use 

tests to be conducted for each fuel cell system type are specified in Table 7.1. 

7.1.2 Steady-state test 

The steady-state test is a test of fuel cell performance at steady inlet fluid temperature and 

flow rate. The steady-state test is conducted at a specified electrical power or thermal 

output rate. The steady-state tests to be conducted for each fuel cell system type are 

specified in Table 7.1. 

7.1.3 Transient electrical power test 

The transient electrical power test is a test of fuel cell performance at steady inlet fluid 

temperature and flow rate and an oscillating electrical power requirement. The transient 

electrical power test is conducted while the electrical load oscillates between minimum and 

maximum power. The transient electrical power test is conducted only for a Type I RFCS. 

7.2 Environmental conditions for testing 

Residential fuel cell system tests shall be conducted at ambient conditions that span the typical 

operating range for the unit. 

Units rated for indoor installation. Residential fuel cell systems rated for indoor installation will 

be tested at ambient conditions of 22 oC (72 oF) and 50 % relative humidity. 

Units rated for outdoor installations. For residential fuel cell systems rated for outdoor 

installation, each set of tests specified in Table 7.1 shall be conducted at ambient conditions of: 

TL = -15 oC (5 oF) and 80 % RH and TH = 35 oC (95 oF) and 50 % RH. 

Table 7.1. Tests required for an outdoor residential fuel cell system type. 

Test Grid 

Independent 

Electrical 

Load 

Following 

(Type I) 

Grid 

Connected 

Constant 

Power 

(Type II) 

Grid 

Connected 

Thermal 

Load 

Following 

(Type III) 

Grid 

Connected 

Water 

Heating 

(Type IV) 

 TH TL TH TL TH TL TH TL

Hot water simulated-use test conducted at 

 Maximum power X X 

 Minimum power X X 

 Rated power X X 

 Thermal load following X X X X 

Steady-state test conducted at 

 Maximum power X X 

 Midpoint power X X 

 Minimum power X X 

 Rated power X X 

 Maximum thermal energy delivery X X 

 Midpoint thermal energy delivery X X 

 Minimum thermal energy delivery X X 

Transient: Min to max power X X 
14

8. Test Procedures 

8.1 Hot water simulated-use test 

8.1.1 Operating conditions 

Table 8.1. Operating conditions and permissible variations for hot water simulated-use test. 

Parameter Operating 

Condition 

Maximum 

Permissible 

Variation 

Real power output, kW As specified for test or 

thermal load following 

±2 % (except for thermal 

load following test) 

Ambient temperature As specified for test ±1.7 °C (3°F) 

Ambient relative humidity As specified for test ±5 %RH 

Heating value of fuel ±1 % 

Fuel pressure at fuel cell inlet ±1 % 

Fuel cell thermal loop flow rate As specified in Note 1. ±2 % 

Make-up water temperature 14.4 °C (58 °F) ±1.1 °C (2 °F) 

Cold water temperature 14.4 °C (58 °F) ±1.1 °C (2 °F) 

Preheat tank operating temperature As specified in Note 2. 

Preheat tank high limit 57.2 °C (135 °F) ±2.8 °C (±5 °F) 

Notes: 

1. The water flow rate through the fuel cell circulating loop, Vh

& , shall be set in 

accordance with the manufacturer’s recommendations. In the absence of 

manufacturer’s recommendations, the flow rate shall be adjusted to produce a water 

temperature rise of 11 °C ± 1.1 °C (20 °F ± 2 °F) at the unit’s maximum rated power . 

2. The operating control for the preheat tank temperature shall be set at a temperature 

designated by the RFCS manufacturer and the unit shall operate within the standard 

control range specified by the manufacturer except that the preheat tank temperature 

shall not be exceed the preheat tank high limit temperature indicated in Table 8.1. 

8.1.2 Initialization. 

The test begins following the initiation of normal operation of the fuel cell to maintain the 

mean preheat tank temperature as specified in Table 8.1. After the initial cutout2

, or any 

subsequent cutout, 40.6 L ± 1.9 L (10.7 gal ± 0.5 gal) are drawn at a rate of 0.19 L/s 

± 0.015 L/s (3 gal/min ±0.25 gal/min). Thirty minutes later, 40.6 L ± 1.9 L (10.7 gal 

± 0.5 gal) are drawn at the same rate. 

8.1.3 Water heating simulated-use test procedure 

The simulated-use test is conducted over a 24 h period with an initial 15 h standby period 

followed by 6 draw periods each having a duration of one hour followed by a 3 h standby 

period. The test begins when the tank temperature is restored to the operating condition 

specified in Table 8.1 following the second draw specified in 8.1.2. When the mean tank 

 

2

 Cutout refers to a control operation that reduces or stops the thermal output from the RFCS when the tank 

temperature setpoint is satisfied. 
15

temperature reaches the specified value, data acquisition is initiated. The system operates 

to maintain the mean tank temperature at the specified value for the first 15 h of the test. 

After an elapsed time of 15 h, a volume equal to 40.6 L (10.7 gal) is withdrawn from the 

water heater at a rate of 0.19 L/s ± 0.015 L/s (3 gal/min ± 0.25 gal/min). At elapsed times 

of 16, 17, 18 and 19 h, additional draws are initiated each having the same volume and 

flow rate as the initial draw. At an elapsed time of 20 h, the quantity of the draw shall be 

increased or decreased as necessary such that the total volume of water withdrawn during 

the entire test equals 243.4 L ±3.8 L (64.3 gal ±1.0 gal). The maximum allowable 

deviation from the mean for any single draw shall be 1.9 L (±0.5 gal). 

The system is allowed to continue to operate to maintain the preheat tank at the operating 

temperature until 24 h have elapsed from the start of the test. After an elapsed time of 23 h, 

the RFCS is stopped. If recovery is occurring at an elapsed time of 23 h, operation of the 

RFCS continues until recovery is complete before the RFCS is stopped. 

8.1.4 Test results 

Initial values are recorded for the test data and data is recorded as specified in Table 8.4 

throughout the 24 h test period. 

8.2 Steady-state test 

8.2.1 Operating conditions 

Operating conditions and maximum permissible variations for the steady state test shall be 

as specified in Table 8.2. 

8.2.2 Initialization. 

The test begins following the initiation of normal operation of the fuel cell. Operation is 

maintained until continuous monitoring indicates that variations in the operating conditions 

are within the maximum permissible ranges specified in Table 8.2. 

Table 8.2. Operating conditions and maximum permissible variations for steady state test. 

Parameter Operating Condition Maximum 

Permissible Variation 

Real power output, kW As specified for test ±2 % 

Ambient temperature As specified for test ± 1.7 oC (3 oF) 

Ambient relative humidity As specified for test ±5 %RH 

Heating value of fuel ±1 % 

Fuel pressure at fuel cell inlet ±1 % 

Fuel cell thermal loop inlet temp. 50 oC (122 oF) ± 1.1 oC (2 oF) 

Make-up water temperature 14.4 °C (58 °F) ±1.1 °C (2 °F) 

Fuel cell thermal loop flow rate As specified in Note. ±2 % 

Thermal energy output rate Response ±2 % 

Fuel flow rate Response ±2 % 

Auxiliary power input Response ±2 % 

Note: The water flow rate through the fuel cell circulating loop, Vh

& , shall be adjusted to 

yield a water temperature rise of 11 °C (20 °F) at the maximum rated power of the unit. 
16

8.2.3 Test results 

Test results include all data specified in Table 8.4 recorded at the indicated intervals 

through the duration of the test. The data is analyzed to determine the mean steady-state 

values and the variation of each parameter. 

8.3 Transient electrical power test 

8.3.1 Operating conditions 

Operating conditions and maximum permissible variations for the transient electrical power 

test shall be as specified in Table 8.3. 

8.3.2 Initialization and test procedure 

The test begins following the initiation of normal operation of the fuel cell at maximum 

power. Operation is maintained until continuous monitoring indicates that variations in the 

operating conditions are within the maximum permissible ranges specified in Table 8.2. 

The electrical load is then reduced in a step change (< 1 s) to minimum power. The system 

operates at minimum power for 15 min. The electrical load is then increased in a step 

change (< 1 s) to maximum power. The system operates at maximum power for 15 min. 

The cycle is repeated 8 times for a total test duration of 4 h. 

8.3.3 Test results 

Test results include all data specified in Table 8.4 recorded at the indicated intervals 

through the duration of the test. The data are analyzed to determine the changes in thermal 

output and fuel use associated with transient electrical loads. 

Table 8.3. Operating conditions and maximum permissible variations for steady state test. 

Parameter Operating Condition Maximum Permissible 

Variation 

Real power output, kW See Section 8.3.2. ±2 % at each power 

level 

Ambient temperature As specified for test ± 1.7 oC (3 oF) 

Ambient relative humidity As specified for test ±5 %RH 

Heating value of fuel ±1 % 

Fuel pressure at fuel cell inlet ±1 % 

Fuel cell thermal loop inlet temp. 50 oC (122 oF) ± 1.1 oC (2 oF) 

Make-up water temperature 14.4 °C (58 °F) ±1.1 °C (2 °F) 

Fuel cell thermal loop flow rate As specified in Note. ±2 % 

Thermal energy output rate Response N.A. 

Fuel flow rate Response N.A. 

Auxiliary power input Response N.A. 

Note: The water flow rate through the fuel cell circulating loop, Vh & , shall be adjusted to 

yield a water temperature rise of 11 °C (20 °F) at the maximum rated power of the unit. 
17

8.4 Data acquisition 

Data for each test will be measured and recorded as specified in Table 8.4. 

Table 8.4. Monitoring points and frequencies for simulated-use, steady-state, and transient tests 

Variable Description Measurement Frequency 

 SimulatedUse Test Steady-state and 

Transient Tests

Ein & electrical power supplied to fuel cell system, kW A A

Eout & electrical power supplied by the fuel cell system, kW A A

Eaux & electrical power supplied to auxiliary devices not 

powered by the RFCS, kW 

A A

pa absolute pressure of air surrounding the RFCS, kPa 

(psia) 

A A

pf absolute pressure of fuel gas at flow measurement 

point, kPa (psia) 

A A

RHa relative humidity of the ambient air, % A A 

RHv relative humidity of the exhaust from the RFCS, % A A

Ta temperature of air surrounding the RFCS, oC (oF) A A

Tci temperature of cold water entering the preheat tank, 

oC (oF) 

A N.A.

Tf absolute temperature of fuel gas, K (R) A A

The temperature of circulating fluid at the exit from the 

RFCS, oC (oF) 

A A

Thi temperature of circulating fluid at the inlet to the 

RFCS, oC (oF) 

A A

Tp temperature of water in preheat tank, oC (oF) A N.A.

Tpe temperature of water leaving preheat tank, oC (oF) A N.A.

Tv absolute temperature of the exhaust gas, K (R) A A

Vc

& volumetric flow of cold water into the preheat tank, 

L/s (gal/min.) 

A N.A.

Vf

& volumetric flow rate of fuel gas, L/h (ft3

/h) A A

Vh

& volumetric flow rate of circulating fluid through the 

RFCS, L/s (ft3

/h) 

A A

0 HHV f higher heating value of fuel gas at standard 

conditions (T0

; P0), kJ/L (Btu/ft3) 

A or B A or B

YCO2 volume fraction of CO2 in dry exhaust gas A A

 Measurement frequency legend 

A = measure and record initial value and subsequent values at 30 s intervals. 

B = measure at the beginning and the end of the test procedure and average. 

N.A. = not applicable. 
18

8.5 Safety 

Fuel cell systems typically involve high voltage electrical systems, flammable gases, 

compressed gases, and high temperatures and thus present a number of hazards. 

Performance testing systems and procedures must comply with applicable codes and 

regulations for construction, operation, and worker safety. Applicable codes for system 

installation may include but are not limited to the standards of the NFPA such as: 

NFPA 853: Standard for the Installation of Stationary Fuel Cell Power Plants 

NFPA 70: National Electrical Code (including Article 692: Fuel Cell Systems) 

NFPA 54: National Fuel Gas Code 

as well as the requirements of IEEE Standard 1547 for Interconnecting Distributed 

Resources With Electric Power Systems. 

Test procedures must be conducted by personnel who are properly trained and aware of the 

hazards and necessary safety procedures for working with fuel systems, electrical systems, 

and fuel cell systems. 
19

9. Data Analysis 

9.1 General description 

Each system rated by this procedure is tested to determine the values specified in Table 8.4 for 

each experiment specified in Table 7.1. The data from each experiment are used to calculate 

mass and energy flow rates. The steady state mass and energy flow rates are used to evaluate 

performance at nominal (or rated) conditions. For electrical load following (Type I) and thermal 

load following (Type III) systems, the mass and energy flow rates are also used to establish 

functional relationships for part-load performance that are then used in the rating procedure. The 

mass and energy flow rates from the simulated use tests are used to determine the fuel use and 

the energy provided to a domestic water preheat tank during each hour of the test procedure. The 

following sections describe the procedures for determining the mass and energy flow rates, 

establishing the rated performance, determining functional relationships for part-load 

performance, and evaluating the hourly performance for the simulated use test. 

9.2 Calculation of Mass and Energy Flow Rates 

9.2.1 Fuel use rate 

The rate at which fuel energy is used by the fuel cell system, F& , expressed in kW (Btu/h) is 

determined by: 

0 1

f

f

0

0

f f C

p

p

T

T F& = V& ×HHV × × (9-1) 

where: 

Vf

& is the volumetric flow rate of the fuel, L/h (ft3

/h) 

0 HHV higher heating value of fuel f gas at standard conditions (T0

; P0), kJ/L (Btu/ft3) 

T0

 is standard temperature, 298 K (537 R) 

Tf is the temperature of the fuel gas at the inlet to the RFCS, K (R) 

p0

 is standard pressure, 101.3 kPa (14.7 psia) 

pf is the pressure of the fuel gas at the inlet to the RFCS, kPa absolute (psia) 

C1 is a units conversion factor, C1 = 1 kW⋅h/3600 kJ (1 Btu/Btu) 

9.2.2 Net electrical power 

The net electrical power provided by the fuel cell system, Efc & , in kW is determined by3: 

Efc Eout Ein Eaux & = & − & − & (9-2) 

where: 

 

3

 The quantity Eout Ein & − & in Eq. (9-2) can be measured directly in lieu of measure the components and calculating 

the difference. The power for auxiliary devices external to the fuel cell, Eout Ein & − & must be subtracted to the yield 

the net electrical power. 
20

Eout & is the power supplied by the RFCS, kW 

Ein & is the power supplied to the RFCS, kW 

Eaux & is the power supplied to any devices external to the RFCS that are required for fuel 

cell operation (e.g., external circulating pumps), kW. 

9.2.3 Steady-state thermal output rate 

During the steady-state test, the thermal output rate from the RFCS, Qfc & , in kW (Btu/h) is 

determined by: 

( ) fc h h h The Thi Q = ρ V& c − & (9-3) 

where: 

ρh is the density of the circulating fluid evaluated at the temperature at the flow 

measurement point, kg/L (lbm/ft3

) 

Vh

& volumetric flow rate of circulating fluid through the RFCS, L/s (ft3

/h) 

ch is the specific heat of the circulating fluid evaluated at the average of the fuel cell 

system inlet and outlet temperatures, kJ/kg⋅oC (Btu/lbm⋅oF) 

The is the temperature of the circulating fluid exiting the RFCS, oC (oF) 

Thi is the temperature of the circulating fluid entering the RFCS, oC (oF). 

9.2.4 Simulated-use thermal output rate 

During the simulated-use test, the useful thermal energy provided for water heating by the 

RFCS in combination with the specified preheat tank, Qwh & , in kW (Btu/h) is given by: 

( ) wh c c c Tpe Tci Q = ρ V& c − & (9-4) 

where: 

ρc is the density of water at Tci, kg/L (lbm/ft3

) 

Vc

& is the volumetric flow rate of cold water into the preheat tank L/s (ft3

/h) 

cc is the specific heat of water evaluated at the average of the cold water inlet and 

preheat tank outlet temperatures, kJ/kg⋅oC (Btu/lbm⋅oF) 

Tpe is the temperature of the water leaving the preheat tank, oC (oF) 

Tci is the temperature of cold water entering the preheat tank, oC (oF). 

9.2.5 Thermal loss rate 

During the simulated-use test, the thermal energy lost to the environment is determined by: 

Qloss Qfc Qwh & & & = − (9-5) 

where Qwh & and Qfc & are calculated as shown above. 
21

9.2.6 Ventilation airflow rate 

For a RFCS that uses indoor air, the air supplied to the fuel cell system may be extracted 

from the house. If the air is extracted from the indoors, it must be replenished by outdoor 

air, and the RFCS output must be corrected to account for the energy to condition the 

outdoor air. For units that use indoor air for ventilation, the rate at which air is supplied to 

the RFCS, is determined based on the flow rate of fuel, the composition of the fuel gas, and 

the mole fraction of carbon dioxide in a dry sample of the exhaust from the RFCS. In 

addition, since the RFCS may operate intermittently, the exhaust flow rate is monitored 

directly to determine the airflow during pre- and post-purge cycles. 

When carbon dioxide is present in the exhaust stream at concentrations exceeding the 

ambient value (indicating that fuel is reacting in the RFCS) the mass basis air-to-fuel ratio, 

raf, is given by: 





 





+ − + +

+ ω = r s 2t

Y

4

12r s 16t

7.2127 7.2424

r dry

CO2

a

af (9-6) 

where: 

dry YCO2 is the volume fraction of CO2 in dry exhaust gas 

r,s,t are determined by the fuel composition expressed as CrHsOt

ωa is the humidity ratio of the ambient air, kg-water/kg-dry air (lbm-water/lbm-dry air): 

( )

( ) a a s a

a s a

a p RH p T

RH p T

− ×

× × = 0.622

ω (9-7) 

where: 

pa is the pressure of the ambient air surrounding the RFCS, kPa-absolute (psia) 

Ta is the absolute temperature ambient air surrounding the RFCS , K (R) 

ps(Ta) is the saturation pressure of water at Ta, kPa (psia) 

The airflow rate into the RFCS in kg/h (lbm/h) can then be found based on the fuel flow: 

V if Y Y ; else 0

T

T

p

p m r 0

CO2

dry

f CO2

f

0

0

0 f

af f

on & a = ρ & > (9-8) 

where variables are as previously defined and: 

0 ρf is the density of the fuel at standard conditions (T0

; p0), kg/L (lbm/ft3) 

When the concentration of carbon dioxide in the exhaust gas does not exceed the ambient 

value, the exhaust is assumed to be comprised solely of air and the mass flow rate of air is 

determined from: 

V if Y Y ; else 0

T

T

p

p m 0

CO2

dry

v CO2

v

0

0

0 v

a

off & a = ρ & ≤ (9-9) 

where variables are as previously defined and: 

0 ρa is the density of air at standard conditions (T0

; P0), kg/L (lbm/ft3) 
22

Vv

& is the volumetric flow rate of the exhaust gas, L/h (ft3

/h) 

Steady-state ventilation airflow rate

The rate at which ventilation air is supplied to the RFCS during the steady state test 

expressed in L/h (ft3

/h) is given by: 

( ) ∫

=

= + − = f

i

t t

t t

off

a

on

a

f i

a m m dt

t t

1

m& & & (9-10) 

where ti and tf are the initial and final times of the steady state test measured in hours. 

Simulated-use ventilation airflow 

The ventilation air supplied to the RFCS during hour i of the simulated use test expressed 

in L (ft3

) is given by: 

( ) ∫

=

= −

= + i

i 1

t t

t t

off

a

on ma,i m& a m& dt (9-11) 

where ti-1 and ti and are the initial and final times, respectively, of hour i expressed in hours. 

9.2.7 Water use 

The rate at which water is supplied to the RFCS, is determined based on the flow rate of 

fuel, the composition of the fuel gas, the relative humidity of the inlet air, the mole fraction 

of carbon dioxide in a dry sample of the exhaust from the RFCS, and the relative humidity 

of the exhaust gas. In addition, since the RFCS may operate intermittently, the exhaust 

flow rate is monitored directly (along with the inlet and exhaust relative humidity values) 

to determine the water added to the air stream during pre- and post-purge cycles. 

When carbon dioxide is present in the exhaust stream at concentrations exceeding the 

ambient value (indicating that fuel is reacting in the RFCS) the mass basis water-to-fuel 

ratio, rwf, is given by: 

( )

( ) ( ) ( )( ) 12r s 16t

18.015

r 0.4020 0.5 s 0.8041 t

Y

1.6082

Y p p T RH

p T RH

r dry a a

CO2

a

v s v v

dry

CO2

s v v

wf

+ +

×



 



 − ω + + ω 













 ω − − ×

× =

 (9-12) 

where variables are as previously defined and: 

ps(Tv) is the saturation pressure of water at Tv, kPa (psia) 

ωv is the humidity ratio of the exhaust air, kg-water/kg-dry air (lbm-water/lbm-dry air) 

RHv is the relative humidity of the exhaust air 

The water flow rate into the RFCS in L/h (gal/h) can then be found based on the fuel flow 

rate: 

V if Y Y ; else 0

T

T

p

p V r 0

CO2

dry

f CO2

f

0

0

f

w

0

f

wf

on

w >

ρ

ρ & = & (9-13) 

where variables are as previously defined and 
23

ρw is the density of water at standard conditions, kg/L (lbm/gal) 

When the concentration of carbon dioxide in the exhaust gas does not exceed the ambient 

value, the exhaust is assumed to be comprised solely of air and the volumetric flow rate of 

water supplied to the RFCS in L/h (gal/h) is determined from: 

V ( ) if Y Y ; else 0

T

T

p

p V 0

CO2

dry

v v a CO2

v

0

0

v

w

0

off a

w ω −ω ≤

ρ

ρ & = & (9-14) 

where variables are as previously defined and: 

ωa is the humidity ratio of the inlet air, kg-water/kg-dry air (lbm-water/lbm-dry air) 

Steady-state water use rate

The rate at which water is used by the RFCS during the steady state test expressed in L/h 

(gal/h) is given by: 

( ) ∫

=

= + − = f

i

t t

t t

off

w

on

w

f i

w,ss V V dt

t t

1 V& & & (9-15) 

where ti and tf are the initial and final times of the steady state test measured in hours. 

Simulated-use water consumption. 

The water used during hour i expressed in L (gal) is given by: 

( ) ∫

=

= −

= + i

i 1

t t

t t

off

w

on Vw,i Vw V dt & & (9-16) 

where ti-1 and ti and are the initial and final times, respectively, of hour i expressed in hours. 

9.3 Transient Electrical Load Factors 

For systems operating in an electrical load following mode, the fuel use and thermal output 

can be affected by transient electrical loads. Correction factors are determined for the fuel 

use and thermal output based on the results from the transient electrical test. The transient 

electrical load correction factor for the fuel use, Ktf, is given by: 

( ) t2 t1 ss

t

t

tf t t F

Fdt

K

t 2

t1

&

&

− = ∫ (9-17) 

where 

tt1 is the time at the beginning of the transient electrical test, h 

tt2 is the time at the end of the transient electrical test, h 

Fss

& is the average rate of fuel use by the RFCS during the steady state test, kW (Btu/h) 

The transient electrical load correction factor for the thermal output, Ktq, is given by: 

( ) t2 t1 fc,ss

t

t fc

tq t t Q

Q dt

K

t 2

t1

&

&

− = ∫ (9-18) 
24

where variables are as previously defined and: 

Qfc,ss & is the average thermal output by the RFCS during the steady state test, kW (Btu/h) 
25

10. Calculation of System Performance Ratings 

10.1 General Description 

The performance test method is based on the predicted response of the system to typical 

residential space heating loads and to the domestic hot water loads specified within the DOE test 

procedure [6] for domestic water heaters. The predicted response of the system to each load 

condition is based on the results of the tests described previously and on the methodology 

outlined in the following sections for each system type. 

The residential loads are developed for three days representing typical winter, spring/fall, and 

summer days. Predicted results for the electrical energy supplied to the residence, electrical 

energy supplied to the grid, thermal energy supplied for water heating, thermal energy supplied 

for space heating, fuel use, and water use are summed over each standard day and weighted by 

the number of days represented by each standard day. The following expressions define the 

annual energy results: 

Annual electrical energy supplied to the residence (AE) in kW⋅h 

∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

i

24

i 1

B

i

24

i 1

A AE (10-1) 91 Ei 183 E 91 E

Annual electrical energy supplied to the utility grid (AG) in kW⋅h 

∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

i

24

i 1

B

i

24

i 1

A AG (10-2) 91 Gi 183 G 91 G

Annual thermal energy supplied for water heating (AQwh) in kW⋅h (Btu) 

∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

wh,i

24

i 1

B

wh,i

24

i 1

A AQwh 91 Qwh,i 183 Q 91 Q (10-3) 

Annual thermal energy supplied for space heating (AQsh) in kW⋅h (Btu) 

∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

sh,i

24

i 1

B

sh,i

24

i 1

A AQ (10-4) sh 91 Qsh,i 183 Q 91 Q

Annual fuel energy supplied to the fuel cell system (AF) in kW⋅h (Btu) 

∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

i

24

i 1

B

i

24

i 1

A AF (10-5) 91 Fi 183 F 91 F

Annual volume of domestic water supplied to the fuel cell system (AW) in L (gal) 

∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

w,i

24

i 1

B

w,i

24

i 1

A AW 91 Vw,i 183 V 91 V (10-6) 

where the superscripts A, B, and C indicate winter, spring/fall, and summer days, respectively, 

and the values for Ei, Gi, Qwh,i, Qsh,i, Fi, and Vw,i are determined from equations presented in the 

following sections for each residential fuel cell system type. Results are reported in the format 

depicted in Table 10.1. 
26

Table 10.1. Annual energy ratings for residential fuel cell systems 

Service Estimated Annual 

Requirement for 

Residence 

Annual Energy 

Supplied by this Unit 

Electrical energy supplied to the residence kW⋅h AE kW⋅h 

Electrical energy supplied to the grid N.A. AG kW⋅h

Thermal energy for water heating kW⋅h (Btu) AQwh kW⋅h (Btu) 

Thermal energy for space heating kW⋅h (Btu) AQsh kW⋅h (Btu) 

Annual natural gas usage by this unit N.A. AF kW⋅h (therms) 

Annual water usage by this unit N.A. AW L (gal) 

10.2 Rated and annual performance for grid independent, electrical load following Type I 

RFCS 

10.2.1 Rated performance for outdoor Type I RFCS 

The rated performance for an outdoor Type I RFCS is defined by the electrical power 

output, thermal output rate, fuel use rate, and water use rate when the RFCS is operated to 

yield maximum continuous power at an ambient temperature of TH. The rated values for 

the energy and mass flow rates are given by: 

Power in kW: E Efc & R = & (10-7a) 

Coincident thermal output rate in kW (Btu/h): Q Qss & R & = (10-7b) 

Fuel use rate in kW (Btu/h): F F& R = & (10-7c) 

Water use rate in L/h (gal/h): w

R V&

w = V& (10-7d) 

where the energy and mass flow rates are determined as described in Section 9.2 and are 

averaged over the duration of the steady-state test. 

10.2.2 Rated performance for indoor Type I RFCS4

The rated performance for an indoor Type I RFCS is defined by the electrical power 

output, thermal output rate, fuel use rate, ventilation airflow rate (if indoor air is used for 

ventilation), and water use rate when the RFCS is operated to yield maximum power at an 

ambient temperature of TI. The rated values for the energy and mass flow rates are given 

by: 

 

4

 Variables associated with indoor systems are designated with an asterisk to distinguish them from similarly defined 

variables for systems located outdoors. 
27

Power in kW: E Efc

R* & = & (10-8a) 

Coincident thermal output rate in kW (Btu/h): Q Qss

R* & & = (10-8b) 

Fuel use rate in kW (Btu/h): F FR* & = & (10-8c) 

Water use rate in L/h (gal/h): w

R* V&

w = V& (10-8d) 

Ventilation airflow rate (if from indoors) in kg/h (lbm/h): a

R* m& a = m& (10-8e) 

where the energy and mass flow rates are determined as described in Section 9.2 and are 

averaged over the duration of the steady-state test. 

10.2.3 Part-load performance evaluation for a Type I RFCS 

10.2.3.1 Overview of part-load performance evaluation 

The annual rating procedure for a Type I RFCS, described in Section 10.2.5, requires partload factors that relate performance to ambient temperature and electrical part-load ratio, re, 

which is defined as: 

E 1h

E

r R

fc

e × = & (10-9) 

For a Type I RFCS, part-load factors expressed as polynomials that are first-order in 

ambient temperature, Ta, and second-order in part-load ratio, re, are defined for the fuel use 

rate (Xf), thermal output rate (Xq), ventilation airflow rate (Xa), and water use rate (Xw). 

The coefficients for each part-load factor polynomial are determined by linear regression 

using data from the six steady-state tests specified in Table 10.2. For each test the system 

is operated to yield the specified electrical part load ratio, re. The environmental chamber 

is operated to maintain the specified ambient temperature which is TH or TL for outdoor 

units and TI for indoor units. Based on the recorded data, results are calculated for the 

power, thermal output rate, fuel flow rate, ventilation airflow rate, and water use rate as 

specified in Section 9.2. These quantities along with ambient temperature are averaged 

over the duration of the steady-state test. Based on these results and the rated values, the 

average electrical part-load ratio and the part-load factors for the fuel use rate, thermal 

output rate ventilation airflow rate, and water use rate are determined. The procedure is 

illustrated in Table 10.2. Data from the final six columns of Table 10.2 are used in the 

linear regressions to determine the coefficients for the polynomial functions. The 

definition, functional form, and application of each part-load factor are presented in the text 

following the table. 

In addition, transient electrical load correction factors for fuel use, Ktf, and thermal output, 

Ktq, are determined in accordance with Eqs. 9-18 and 9-19, respectively, at TH and TL for 

outdoor units and at TI for indoor units. 
28

Table 10.2. Procedure for determining part-load factors for energy and mass flow rates as 

functions of electrical part-load ratio and ambient temperature. 

Test conditions Calculated results 

 re Ta Mass and energy flows1 Part load factors Independent 

variables 

 Efc & Qfc & F& ma & 3 Vw

& Xf Xq Xw Xa re Ta

1.0 TH 

Midpt TH

Min2 TH Eq. Eq. Eq. N.A. Eq. Eq. Eq. Eq. N.A. Note 1 Note 1

1.0 TL (9-2) (9-3) (9-1) (9-15) (10-10) (10-13) (10-16) 

Midpt TL 

Outdoor 

RFCS 

Min2 TL 

1.0 TI 

Midpt TI Note 1 Note 1

Indoor 

RFCS Min2

 TI

Eq. 

(9-2) 

Eq. 

(9-3) 

Eq. 

(9-1)

Eq.

(9-10)

Eq. 

(9-15)

Eq. 

(10-10)

Eq. 

(10-13)

Eq. 

(10-16)

Eq. 

(10-22) 

Notes: 1. Quantities are averaged over the duration of the steady state test. 

 2. Minimum fuel cell output is defined by the manufacturer. 

 3. Ventilation airflow rate is determined for units that draw air from indoors. 

10.2.3.2 Part-load factors for outdoor Type I RFCS 

Part-load fuel use factor. The part-load fuel use factor, Xf, is the fraction of the rated fuel 

use rate at the conditions of electrical part load ratio, re, and system ambient temperature 

Ta: 

( )

R

e a

f F

F r ,T X &

& ≡ (10-10) 

The functional form of Xf is: 

( ) a 6 a

2

4 e a 5 e

2 Xf re ,Ta = a1 + a 2re + a 3re + a r T + a r T + a T (10-11) 

where the coefficients are determined by linear regression of the steady state test data. 

Representative functions for the part-load fuel use factor are illustrated in Figure 10.1. 

The part-load fuel use factor is used in the annual performance rating procedure to 

determine the fuel use, F& , in kW (Btu/h) as a function of the electrical power and ambient 

temperature according to: 

( ) R F& = Xf re ,Ta ×F& (10-12) 
29

TL, 0.2, 0.3

TL, 0.6, 

0.62

TL, 1, 1.08

TH, 0.2, 0.2

TH, 0.6, 

0.56

TH, 1, 1

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Electrical part-load ratio

Part-load fuel use factor

TL

TH

TL, 1, 0.9

TL, 0.6, 0.5

TL, 0.2, 0.2

TH, 1, 1

TH, 0.6, 0.7

TH, 0.2, 0.4

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Electrical part-load ratio

Part-load thermal output factor

TL

TH

( ) a 6 a

2

4 e a 5 e

2 Xf re ,Ta = a1 + a 2re + a 3re + a r T + a r T + a T ( ) a 6 a

2

4 e a 5 e

2 Xq re ,Ta = b1 + b2re + b3re + b r T + b r T + b T

a1 = 0.2994 a2 = 0.2625 a3 = 0.5781 b1 = -0.03438 b2 = 0.3125 b3 = 0.5469 

a4 = 0.0125 a5 = -0.00938 a6 = -0.00713 b4 = 0.0125 b5 = -0.01563 b6 = 0.008125 

Figure 10.1. Representative function for Figure 10.2. Representative function for 

 part-load fuel use factor part-load thermal output factor 

Part-load thermal output factor. The part-load thermal output factor, Xq, is the fraction of 

the rated thermal output rate at the conditions of electrical part load ratio, re, and system 

ambient temperature Ta: 

( )

R

fc e a

q Q

Q r ,T X &

& ≡ (10-13) 

The functional form of Xq is: 

( ) a 6 a

2

4 e a 5 e

2 Xq re ,Ta = b1 + b2re + b3re + b r T + b r T + b T (10-14) 

where the coefficients are determined by linear regression of the steady state test data. 

Representative functions for the part-load thermal output factor are illustrated in Figure 

10.2. 

The part-load thermal output factor is used in the annual performance rating procedure to 

determine the thermal output rate, Qfc & , in kW (Btu/h) as a function of the electrical power 

and ambient temperature according to: 

( ) R Q& fc Xq re ,Ta Q& = (10-15) 

Part-load water use factor. The part-load water use factor, Xw, is the fraction of the rated 

water flow rate required at the conditions of electrical part load ratio, re, and system 

ambient temperature Ta: 

( )

R

w

w e a

w V

V r ,T X &

& ≡ (10-16) 
30

The functional form of Xw is: 

( ) a 6 a

2

4 e a 5 e

2 Xw re ,Ta = c1 + c2re + c3re + c r T + c r T + c T (10-17) 

where the coefficients are determined by linear regression of the steady state test data. 

Representative functions for the part-load thermal output factor are illustrated in Figure 

10.3. 

The part-load water use factor is used in the annual performance rating procedure to 

determine the water use rate, Vw

& , in L/h (gal/h) as a function of the electrical power and 

ambient temperature according to: 

( ) R

w w e Ta Vfw V X r , & = & (10-18) 

10.2.3.3 Part-load factors for indoor Type I RFCS 

The part-load factors for an indoor Type I RFCS are defined and applied in the same way 

as those for outdoor units except there is no functional dependence on ambient temperature 

because it is assumed to remain constant at a value of TI. Thus the forms for the part-load 

functions are simplified to: 

( ) 2

e

*

e 3

*

2

*

e 1

*

f X r = a + a r + a r (10-19) 

( ) 2

e

*

e 3

*

2

*

e 1

*

q X r = b + b r + b r (10-20) 

( ) 2

e

*

e 3

*

2

*

e 1

*

w X r = c + c r + c r (10-21) 

In addition, if the RFCS draws ventilation air from the residence, it imposes air 

conditioning and heating loads. To model these loads, a part-load ventilation airflow factor 

is defined for units that draw ventilation air from the residence. The part-load ventilation 

airflow factor, Xa, is the fraction of the rated ventilation airflow required at the conditions 

of electrical part load ratio, re, and system ambient temperature TI: 

( )

R*

a

e

*

* a

a m

m r X &

& ≡ for Ta = TI (10-22) 

The functional form of Xa is: 

( ) 2

e

*

e 3

*

2

*

e 1

*

a X r = d + d r + d r (10-23) 

where the coefficients are determined by linear regression of the steady state test data. 

Representative functions for the part-load thermal output factor are illustrated in Figure 

10.4. 

The part-load ventilation airflow factor is used in the annual performance rating procedure 

to determine the ventilation airflow rate, * ma & , in kg/h (lbm/h) as a function of the electrical 

part-load ratio: 

( ) R*

e a

*

a

* m& a = X r m& (10-24) 
31

TL, 1, 1.1

TL, 0.6, 

0.65

TL, 0.2, 

0.33

TH, 1, 1

TH, 0.6, 

0.57

TH, 0.2, 

0.23

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Electrical part-load ratio

Part-load water use factor

TL

TH

TI, 1, 1

TI, 0.6, 

0.5975

TI, 0.2, 

0.2645

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Electrical part-load ratio

Part-load ventilation airflow factor

TI

( ) a 6 a

2

4 e a 5 e

2 Xf re ,Ta = c1 + c2re + c3re + c r T + c r T + c T ( ) 2

e

*

e 3

*

2

*

e 1

*

a X r = d + d r + d r

c1 = 0.3125 c2 = 0.3625 c3 = 0.5000 * d1 = 0.1241 * d2 = 0.6588 *

3 d = 0.2172 

c4 = 0.00750 c5 = -0.00625 c6 = -0.00625 

Figure 10.3. Representative function for Figure 10.4. Representative function for 

 part-load water use factor part-load ventilation airflow 

 factor 

10.2.4 Analysis of Simulated Use Data for Type I RFCS 

10.2.4.1 Hot water simulated use data for outdoor Type I RFCS 

For an outdoor Type I RFCS, the thermal energy provided for water heating is a function of 

the hot water load, the electrical part load ratio, and the outdoor temperature. Thus, the hot 

water simulated use tests are conducted at combinations of ambient temperature and 

electrical part load ratio as specified in Table 7.1. The thermal energy for water heating for 

each hour, Qwh,i, in kW⋅h (Btu) is found from: 

Q Q ( ) r ,T dt e a

t t

t t wh,i wh

i

i 1 ∫

=

= −

= & (10-25) 

where: 

Qwh & is the thermal output rate evaluated from Eq. 9-4 at the indicated values of 

power and ambient temperature, kW (Btu/h) 

Using this data, expressions are determined for the thermal energy provided for each hour 

as a function of the electrical part load ratio and the ambient temperature: 

wh,i 1i 2i a 3i e 4i eTa Q = u + u T + u r + u r (10-26) 
32

where the coefficients u1i, u2i, u3i, and u4i are determined by linear regression for each 

hour i of the simulated-use test. 

The average thermal loss, Qloss,L & and Qloss,H & , are quantities of thermal power produced by 

the fuel cell unit during the hot water simulated use tests at ambient temperatures TL and 

TH, respectively, that cannot be supplied as useful heat. They are calculated over the hot 

water simulated use test as: 

Q ( ) T dt

τ

1 Q L

 t t

 t t loss,L loss

i

i 1 ∫

=

= −

= & & (10-27) 

Q ( ) T dt

τ

1 Q H

 t t

 t t loss,H loss

i

i 1 ∫

=

= −

= & & (10-28) 

where: 

Qloss & is the thermal loss rate evaluated from Eq. 9-5 at the indicated value of ambient 

temperature. 

10.2.4.2 Hot water simulated use data for indoor Type I RFCS 

For an indoor Type I RFCS, the thermal energy provided for water heating is a function of 

the hot water load and the electrical part load ratio with the ambient temperature remaining 

constant at a value of TI. Thus, the hot water simulated use tests are conducted at ambient 

temperature TI and at values of electrical part load ratio as specified in Table 7.1. The 

thermal energy for water heating for each hour, Qwh,i, in kW⋅h (Btu) is found from: 

Q Q ( ) r ,T dt e I

t t

t t wh

*

wh,i

i

i 1 ∫

=

= −

= & (10-29) 

where: 

Qwh & is the thermal output rate evaluated from Eq. 9-4 at the indicated values of 

power and ambient temperature, kW (Btu/h) 

Using this data, expressions are determined for the thermal energy provided for each hour 

as a function of the electrical part load ratio: 

e

*

2i

*

1i

*

wh,i Q = u + u r (10-30) 

where the coefficients * u and 1i

* u are determined for each hour i of the hot water 2i

simulated-use test. 

The average thermal loss, * Qloss & , in kW (Btu/h) is thermal power produced by the fuel cell 

unit that cannot be supplied as useful heat. It is calculated over the hot water simulated use 

test as: 

Q ( ) T dt

τ

1 Q I

 t t

 t t loss

*

loss

i

i 1 ∫

=

= −

= & & (10-31) 

where: 
33

Qloss & is the thermal loss rate evaluated from Eq. 9-5 at the indoor ambient 

temperature, TI. 

10.2.5 Annual Performance Rating Procedure for Type I RFCS 

10.2.5.1 Overview of rating procedure 

Using the procedure described in Section 10.2.2, steady-state test results can be used to 

describe the fuel use rate, thermal output rate, ventilation airflow rate, and water used rate 

of a RFCS as functions of the electrical part load ratio and the ambient temperature. The 

approach described in Section 10.2.3 can be used to establish functions that describe the 

performance of a RFCS (in combination with a preheat tank) during each hour of a hot 

water simulated use test. This section describes how the steady-state functions and the 

hour-by-hour results for the hot water simulated use test can be combined to simulate the 

annual performance of a Type I RFCS. 

10.2.5.2 Rating procedure for outdoor Type I RFCS 

The net electrical energy supplied by the RFCS, Ei, in kW⋅h is given by: 

( ) L,i

R Ei = min E& ×1h,E (10-32) 

where R E& was defined previously and: 

E is the electrical energy requirement L,i of the residence during hour i, kW⋅h 

The electrical energy output is used to determine an electrical part load ratio, re,i: 

E 1h

E

r R

i

e,i

× = & (10-33) 

The fuel use, thermal energy supplied by the fuel cell, thermal energy applied to water 

heating, and water use are determined by this electrical part load ratio, the ambient 

temperature for the hour, Tai, and the appropriate transient electrical load correction factors: 

( ) ( )( ) () K () () T K T K T 1h

T T

T T

F X r ,T F tf H tf L tf L

H L

R a,i L

i f e,i a,i  ×



 



 − + −

− = × & × (10-34) 

( ) 1 h

T T

T T

Q Q Q Q

L H

a,i H

loss,i loss,H loss,L loss,H  ×



 





−

− = + − × & & & (10-35) 

( ) ( )( ) K () () T K T K ( ) T 1h

T T

T T

Q X r ,T Q tq H tq L tq L

H L

R a,i L

fc,i q e,i a,i  ×



 



 − + −

− = × × & (10-36) 

wh,i 1i 2i a,i 3i e,i 4i e,i

Ta,i Q = u + u T + u r + u r (10-37) 

V X ( ) r ,T V 1h R

w = w e,i a,i × &fw × (10-38) 

where variables are as previously defined and: 
34

Fi is the fuel use during hour i, kW⋅h (Btu) 

Qfc,i the thermal energy provided by the fuel cell for water heating and space heating 

during hour i, kW⋅h (Btu) 

Qwh,i the thermal energy provided for water heating during hour i, kW⋅h (Btu) 

V total volume of water used for hour i , L (gal) w,i

The thermal energy used for heating water and the thermal loss between the fuel cell and 

the preheat tank are subtracted from the RFCS thermal output, Qfc,i, to determine the 

thermal energy available for space heating, Qav,i in kW⋅h (Btu): 

Qav,i = Qfc,i − Qwh,i − Qloss,i (10-39) 

The net contribution of the RFCS to space heating is the minimum of the available thermal 

energy and the space heating load, QLs,i in kW⋅h (Btu): 

( ) sh,i av,i QLs,i Q = min Q , (10-40) 

Annual results are summed over each representative day, weighted by the number of 

occurrences of each representative day, and reported in the format described in Section 

10.1. 

10.2.5.3 Rating procedure for indoor Type I RFCS 

For indoor Type I RFCS, a first approximation to the electrical energy supplied by the fuel 

cell for each hour, E1,i, in kW⋅h is the minimum of the fuel cell rated power applied over 

the entire hour or the electrical energy requirement of the residence: 

( ) L,i

R E1,i = min E& ×1h,E (10-41) 

If the RFCS uses indoor air for ventilation, this first approximation must be corrected to 

reflect the increase in electrical energy required by the residence due to cooling and 

dehumidification of the ventilation air required by the RFCS. The quantity of air is 

determined by the first approximation to the part load ratio, r1e,i: 

E 1h

E

r R

1,i

1e,i

× = & (10-42) 

which is used in conjunction with the ambient temperature for the hour, Tai, to determine 

the mass of ventilation air, ma,i, in kg (lbm) according to: 

m X ( ) r m 1h R*

a 1e,i a

*

a,i = × & × (10-43) 

The electrical energy required to condition this air, * ∆Ea,i , in kJ (Btu) is calculated by: 

[ ] ( ) 















∆ = − + ω −ω × ;0

COP

C E max m h h h

clg

1

oa iac cd oa iac

*

a,i

*

a,i (10-44) 

where: 
35

hoa is the enthalpy per unit mass of dry air (da) for the outdoor air at hour i, kJ/kg 

dry air (Btu/lbm dry air) 

hiac is the enthalpy per unit mass of dry air for the indoor air at cooling season 

conditions of 25 oC (77 oF) and 50 % RH, 50.3 kJ/kg dry air (21.6 Btu/lbm dry 

air) 

hcd is the enthalpy of the condensate at 25 oC (77 oF), 104.81 kJ/kg (45.1 Btu/lbm) 

ωoa is the humidity ratio of the for the outdoor air at hour i, kg-water/kg-dry air 

(lbm water/lbm dry air) 

ωiac is the humidity ratio of the indoor air at cooling season conditions of 25 oC 

(77 oF) and 50 % RH, 0.00988 kg-water/kg-dry air (0.00988 lbm water/lbm dry 

air) 

COPclg is the coefficient of performance for the air conditioning system which is 

assumed to be 3.5 (corresponding to an EER of 12) 

C1 is a constant for units conversion, 1 kW⋅h/3600 kJ (kW⋅h/3413 Btu) 

The net electrical energy supplied by the indoor RFCS, * E , in kW i ⋅h is given by: 

[( ) ] L,i

*

a,i

* R* Ei = min E& ×1h − ∆E ,E (10-45) 

where the correction, * ∆Ea,i , is zero if the unit does not draw air from the indoors. The 

remaining mass and energy flows are based on the gross electrical energy output of the 

indoor RFCS, * Egross,i in kW⋅h: 

*

a,i

*

i

* Egross,i = E + ∆E (10-46) 

This gross electrical energy output is used to determine an electrical part load ratio, *

e,i r : 

E 1h

E

r R*

*

* gross,i

e,i

× = & (10-47) 

The fuel use, thermal energy supplied by the fuel cell, thermal energy applied to water 

heating, and water use are determined by this electrical part load ratio and the appropriate 

transient electrical load correction factors: 

F X (r ) F K 1h tf

* R*

e,i

*

f

*

i = × & × × (10-48) 

Q X (r ) Q K 1h tq

* R*

e,i

*

q

*

fc,i = × × × & (10-49) 

*

e,i

*

2i

*

1i

*

wh,i Q = u + u r (10-50) 

V X (r ) V 1h R*

fw

*

e,i

*

w

*

w = × & × (10-51) 

If the indoor RFCS uses ventilation air from indoors, the thermal output must be corrected 

to reflect the increase in the thermal load of the residence due to the ventilation air required 
36

by the RFCS. The thermal energy required to heat this air, * ∆Qa,i , in kJ (Btu) is calculated 

by: 

∆Q max[m c ( ) T T ;0] pa iah oa

*

a,i

*

a,i = − (10-52) 

where the mass of air for hour i, * ma,i , is determined from Eq. 10-43 and: 

cpa is the constant pressure specific heat for air at assumed conditions of 5 oC 

(40oF), 

Toa is the temperature of the outdoor air at hour i, oC (oF) 

Tiah is the temperature of the indoor air at heating conditions which is 21 oC (70 oF) 

The thermal energy used for heating water, used for heating ventilation air, and lost in the 

circulating fluid between the fuel cell and the preheat tank are subtracted from the RFCS 

thermal output, * Qfc,i , to determine the thermal energy available for space heating, * Qav,i , in 

kW⋅h (Btu)5

: 

loss

*

ah,i

*

wh,i

*

fc,i

* Qav,i = Q − Q − ∆Q − Q (10-53) 

where the correction, * ∆Qa,i , is zero if the unit does not draw air from the indoors. The net 

contribution of the RFCS to space heating is the minimum of the available thermal energy 

and the space heating load: 

( ) Ls,i

*

av,i

* Qsh,i = min Q ,Q (10-54) 

Annual results are summed over each representative day, weighted by the number of 

occurrences of each representative day, and reported in the format described in Section 

10.1. 

 

5

 Note that the available thermal energy may actually be negative if the thermal energy remaining after water heating 

is not sufficient to heat the ventilation air. 
37

10.3 Rated and annual performance for grid interconnected, constant power Type II 

RFCS 

10.3.1 Rated performance for outdoor Type II RFCS 

The RFCS is operated to yield the power or thermal output rate specified by the 

manufacturer at an ambient temperature of TH. The rated values for the energy and mass 

flow rates are given by: 

Power in kW: E Efc & R = & (10-55a) 

Coincident thermal output rate in kW (Btu/h): Q Qss & R & = (10-55b) 

Fuel use rate in kW (Btu/h): F F& R = & (10-55c) 

Water use rate in L/h (gal/h): w

R V&

w = V& (10-55d) 

where the energy and mass flow rates are determined as described in Section 9.2 and are 

averaged over a one-hour steady-state test. 

10.3.2 Rated performance for indoor Type II RFCS 

The RFCS is operated to yield the power or thermal output rate specified by the 

manufacturer at an ambient temperature of TI for indoor units. The rated values for the 

energy and mass flow rates are given by: 

Power in kW: E Efc

R* & = & (10-56a) 

Coincident thermal output rate in kW (Btu/h): Q Qss

R* & & = (10-56b) 

Fuel use rate in kW (Btu/h): F FR* & = & (10-56c) 

Water use rate in L/h (gal/h): w

R* V&

w = V& (10-56d) 

Ventilation airflow rate (if from indoors) in kg/h (lbm/h): on

a

R* m& a = m& (10-56e) 

where the energy and mass flow rates are determined as described in Section 9.2 and are 

averaged over a one-hour steady-state test. 

10.3.3 Fuel use, thermal output, ventilation airflow, and water use rates as functions 

of ambient temperature for Type II RFCS 

10.3.3.1 Overview of part-load performance evaluation 

The annual rating procedure, described in Section 10.3.5, requires that the fuel use rate, 

thermal output rate, water use, and ventilation airflow rate (for indoor units drawing air 

from indoors) be estimated at the appropriate ambient temperature (outdoor or indoor 

temperature depending on location of the system). For outdoor units, system performance 

is determined at two temperatures with the performance at other temperatures estimated by 

interpolation. For indoor units the Type II system always performs at rated conditions. 

10.3.3.2 Part-load factors for outdoor Type II RFCS 

For an outdoor Type II RFCS, the annual rating procedure, described in Section 10.3.5, 

requires that the fuel use rate, thermal output rate, and water use rate be determined at 
38

ambient temperatures of TH and TL. The values at a particular ambient temperature are 

then estimated based on interpolation between the results at TH and those at TL. 

Fuel use. The average hourly fuel use values at TH and TL, respectively, are given by: 

F( ) E ,T dt 1 F H

R t

t 0 H & & ∫

=τ

= τ = (10-57a) 

F( ) E ,T dt 1 F L

R t

t 0 L & & ∫

=τ

= τ = (10-57b) 

where: 

τ is the duration of the steady state test, h 

F& is the fuel use rate evaluated from Eq. 9-1 at the indicated values of power and 

ambient temperature, kW (Btu/h) 

FH is the average hourly fuel use at ambient temperature TH, kW⋅h (Btu) 

FL is the average hourly fuel use at ambient temperature TL, kW⋅h (Btu) 

Thermal output. The average hourly thermal output values at TH and TL, respectively, are 

given by: 

Q ( ) E ,T dt 1 Q H

R t

t 0 fc,H fc & & ∫

=τ

= τ = (10-58a) 

Q ( ) E ,T dt 1 Q L

R t

t 0 fc,L fc & & ∫

=τ

= τ = (10-58b) 

where: 

Qfc & is the thermal output rate evaluated from Eq. 9-3 at the indicated values of 

power and ambient temperature, kW (Btu/h) 

Qfc,H is the average hourly thermal output at ambient temperature TH, kW⋅h (Btu) 

Qfc,L is the average hourly thermal output at ambient temperature TL, kW⋅h (Btu) 

Water use. The average hourly water use quantities at TH and TL, respectively, are given 

by: 

V ( ) E ,T dt 1 V H

R t

t 0 w,H w

& & ∫

=τ

= τ = (10-59a) 

V ( ) E ,T dt 1 V L

R t

t 0 w,L w

& & ∫

=τ

= τ = (10-59b) 

where: 

Vw

& is the water use rate from Eq. 9-15 at the indicated values of power and ambient 

temperature, L/h (gal/h) 

Vw,H is the average hourly water use quantity at ambient temperature TH, kg (lbm) 
39

Vw,L is the average hourly water use quantity at ambient temperature TL, kg (lbm) 

10.3.3.3 Indoor Type II RFCS 

For an indoor Type II RFCS, the annual rating procedure, described in Section 10.3.5, 

assumes that the unit always operates at rated power, fuel use rate, thermal output rate, 

ventilation airflow rate (for units drawing air from indoors), and water use rate. 

10.3.4 Analysis of Simulated Use Data for Type II RFCS 

10.3.4.1 Hot water simulated use data for outdoor Type II RFCS 

For an outdoor Type II RFCS, the thermal energy provided by the RFCS is determined for 

each hour of a hot water simulated use test performed at ambient temperatures of TH and 

TL. In the rating procedure of Section 10.4, the values at a particular ambient temperature 

are estimated based on interpolation between the results at TH and those at TL. Hourly 

values in kW⋅h (Btu) for thermal energy provided by the RFCS/preheat tank at TH and TL, 

respectively, are given by: 

Q Q ( ) E ,T dt H

R t t

t t whH,i wh

i

i 1

& & ∫

=

= −

= (10-60a) 

Q Q ( ) E ,T dt L

R t t

t t whL,i wh

i

i 1

& & ∫

=

= −

= (10-60b) 

where: 

Qwh & is the thermal output rate evaluated from Eq. 9-4 at the indicated values of 

power and ambient temperature, kW (Btu/h) 

QwhH,i is the thermal energy provided by the RFCS/preheat tank during hour i at 

ambient temperature TH, kW⋅h (Btu) 

QwhL,i is the thermal energy provided by the RFCS/preheat tank during hour i at 

ambient temperature TL, kW⋅h (Btu) 

The average thermal loss, Qloss,L & and Qloss,H & , are quantities of thermal power produced by 

the fuel cell unit during the hot water simulated use tests at ambient temperatures TL and 

TH, respectively, that cannot be supplied as useful heat. They are calculated over the hot 

water simulated use test as: 

Q ( ) T dt 1 Q L

t t

t t loss,L loss

i

i 1 ∫

=

τ = −

= & & (10-61a) 

Q ( ) T dt 1 Q H

t t

t t loss,H loss

i

i 1 ∫

=

τ = −

= & & (10-61b) 

where: 

Qloss & is the thermal loss rate evaluated from Eq. 9-5 at the indicated value of ambient 

temperature. 
40

10.3.4.2 Hot water simulated use data for indoor Type II RFCS 

The thermal energy provided by an indoor Type II RFCS system to the domestic water 

heating system of the residence during hour i is given by: 

Q Q ( ) E ,T dt I

R t t

t t wh

*

wh,i

i

i 1

& & ∫

=

= −

= (10-62) 

where ( )I

R Q& wh E& ,T is determined in accordance with Eq. (9-4) using data from a hot water 

simulated use test conducted at rated power and at ambient temperature TI. 

The average thermal loss, * Qloss & , in kW (Btu/h) is thermal power produced by the fuel cell 

unit that cannot be supplied as useful heat. It is calculated over the hot water simulated use 

test as: 

Q ( ) T dt 1 Q I

t t

t t loss

*

loss

i

i 1 ∫

=

τ = −

= & & (10-63) 

Where Qloss & is the thermal loss rate evaluated from Eq. 9-5 at the indoor ambient temperature, TI. 

10.3.5 Annual Performance Rating Procedure 

10.3.5.1 Overview of rating procedure 

Using the procedure described in the Section 10.3.2, steady-state test results can be used to 

determine average hourly values for the fuel use rate, thermal output rate, ventilation 

airflow rate, and water used rate of a Type II RFCS at specific values of the ambient 

temperature (TI for indoor units and both TH and TL for outdoor units). The approach 

described in Section 10.3.3 can be used to determine the performance of a Type II RFCS 

(in combination with a preheat tank) during each hour of a hot water simulated use test at 

specific values of the ambient temperature (TI for indoor units and both TH and TL for 

outdoor units). This section describes how the steady-state functions and the hour-by-hour 

results for the hot water simulated use test can be combined to determine the annual 

performance of a Type II RFCS. 

10.3.5.2 Rating procedure for outdoor Type II RFCS 

In grid interconnected, constant power mode, the electrical energy supplied by the Type II 

RFCS for each hour, i, is always the same and corresponds to the rated power of the fuel 

cell applied over the entire hour. The portion of the net electrical energy that is used by the 

residence, Ei, in kW⋅h is the minimum of the net energy available from the RFCS operating 

at rated power, R E& , and the electrical energy requirement of the residence: 

( ) EL,i = min ×1h, R Ei E& (10-64) 

The remaining power (if any), Gi, in kW⋅h is supplied to the electrical grid: 

i

R Gi = E& ×1h − E (10-65) 

For outdoor units, the fuel use, thermal energy supplied by the fuel cell, thermal energy 

applied to water heating, and water use are determined by using the ambient temperature 

for each hour, Ta,i, to interpolate between the results from the steady-state and simulated 
41

use tests conducted at high (TH) and low (TL) ambient temperatures and at the rated power 

output: 

( )

L H

a,i H

i H L H T T

T T

F F F F −

− = + − × (10-66) 

( ) 1 h

T T

T T

Q Q Q Q

L H

a,i H

loss,i loss,H loss,L loss,H × −

− = + − × & & & (10-67) 

( )

L H

a,i H

fc,i fc,H fc,L fc,H T T

T T

Q Q Q Q −

− = + − × (10-68) 

( )

L H

a,i H

wh,i wh,H wh,L wh,H T T

T T

Q Q Q Q −

− = + − × (10-69) 

( )

L H

a,i H

w,i w,H w,L w,H T T

T T

V V V V −

− = + − × (10-70) 

The thermal energy used for heating water is subtracted from the rated RFCS thermal 

output, Qfc,i, to determine the thermal energy available for space heating, Qav,i, in kWh 

(Btu):

Qav,i = Qfc,i − Qwh,i − Qloss,i (10-71) 

The net contribution of an outdoor Type II RFCS to space heating, Qsh,i, in kW⋅h (Btu) is 

the minimum of the available thermal energy and the space heating load: 

( ) sh,i av,i QLs,i Q = min Q , (10-72) 

Annual results are summed over each representative day, weighted by the number of 

occurrences of each representative day, and reported in the format described in Section 

10.1. 

10.3.5.3 Rating procedure for indoor Type II RFCS 

In grid interconnected, constant power mode, the electrical energy supplied by the Type II 

RFCS for each hour, i, is always the same and corresponds to the rated power of the fuel 

cell applied over the entire hour. For units drawing ventilation air from indoors, the net 

electrical energy must be reduced to reflect the energy used to condition the ventilation air. 

The ventilation airflow rate at rated power, R ma & , is determined during the rating test. The 

electrical energy required to condition this air, ∆Ea,i, in kJ (Btu) is calculated by: 

[ ] ( ) 















∆ = × × − + ω − ω × ;0

COP

C E max m 1 h h h h

c lg

1

oa iac cd oa iac

R*

a

*

a,i & (10-73) 

where: 

hoa is the enthalpy per unit mass of dry air (da) for the outdoor air at hour i, kJ/kgda (Btu/lbm-da) 
42

hiac is the enthalpy per unit mass of dry air for the indoor air at cooling season 

conditions of 25 oC (77 oF) and 50 % RH, 50.3 kJ/kg dry air (21.6 Btu/lbm dry 

air) 

hcd is the enthalpy of the condensate at 25 oC (77 oF), 104.81 kJ/kg (45.1 Btu/lbm) 

ωoa is the humidity ratio of the for the outdoor air at hour i, kg-water/kg dry air (lbm

water/lbm dry air) 

ωiac is the humidity ratio of the indoor air at cooling season conditions of 25 oC 

(77oF) and 50 % RH, 0.00988 kg-water/kg dry air (0.00988 lbm water/lbm dry 

air) 

COPclg is the coefficient of performance for the air conditioning system which is 

assumed to be 3.5 (corresponding to an EER of 12) 

C1 is a constant for units conversion, 1 kW⋅h/3600 kJ (kWh/3413 Btu) 

The portion of the net electrical energy that is used by the residence is the minimum of the 

net energy available from the RFCS (after correcting for the electrical energy to condition 

ventilation air) and the electrical energy requirement of the residence: 

[( ) ] L,i

*

a,i

* R* Ei = min E& ×1h − ∆E ,E (10-74) 

where the correction, ∆Eai, is zero if the unit does not draw air from the indoors. The 

remaining electrical energy (if any), * G , in kW i ⋅h is supplied to the electrical grid: 

( ) * *

,

* * 1h a i i

R Gi = E& × − ∆E − E (10-75) 

For indoor units, the fuel use, thermal energy supplied by the fuel cell, thermal energy and 

water use correspond to the rated values: 

F F 1h * R*

i = & × (10-76) 

Q Q 1h * R*

fc,i = × & (10-77) 

1h * *

, = × R Vw i Vw

& (10-78) 

The thermal energy applied to water heating for each hour corresponds of the values 

obtained from the simulated use test conducted at ambient temperature TI. 

*

wh,i

* Qwh,i = Q from Eq. 10-62 (10-79) 

For indoor units that draw ventilation air from the indoors, the available thermal energy 

must be reduced by the amount of thermal energy required for heating ventilation air, ∆Qa,i, 

in kJ (Btu) which is calculated by: 

Q max[m 1h c ( ) T T ;0] pa iah oa

R*

a

* ∆ a,i = & × × − (10-80) 

where: 

cpa is the constant pressure specific heat for air at assumed conditions of 5 oC 

(40 oF), 
43

Toa is the temperature of the outdoor air at hour i, oC (oF) 

Tiah is the temperature of the indoor air at heating conditions which is 21 oC (70 oF) 

The thermal energy used for heating water and for heating ventilation air is subtracted from 

the rated RFCS thermal output, Qfc,i, to determine the thermal energy available for space 

heating, * Qav,i , in kW⋅h (Btu):

fc,i wh,i ah,i loss,i

* Qav,i = Q − Q − ∆Q − Q (10-81) 

where the correction term , ∆Qa,i, is zero if the unit doe not draw air from the indoors. The 

net contribution of a Type II RFCS to space heating, * Qsh,i , in kW⋅h (Btu) is the minimum 

of the available thermal energy and the space heating load: 

( ) Ls,i

*

av,i

* Qsh,i = min Q ,Q (10-82) 

Annual results are summed over each representative day, weighted by the number of 

occurrences of each representative day, and reported in the format described in Section 

10.1. 
44

10.4 Rated and annual performance for grid interconnected, thermal load following Type 

III RFCS 

10.4.1 Rated performance for outdoor Type III RFCS 

The RFCS is operated to yield maximum thermal output at an ambient temperature of TH. 

The rated values for the energy and mass flow rates are given by: 

Coincident power in kW: E Efc & R = & (10-83a) 

Thermal output rate in kW (Btu/h): Q Qss & R & = (10-83b) 

Fuel use rate in kW (Btu/h): F F& R = & (10-83c) 

Water use rate in L/h (gal/h): w

R V&

w = V& (10-83d) 

where the energy and mass flow rates are determined as described in Section 9.2 and are 

averaged over the duration of the steady-state test. 

10.4.2 Rated performance for Type III RFCS 

The RFCS is operated to yield maximum thermal output at an ambient temperature of TI

for indoor units. The rated values for the energy and mass flow rates are given by: 

Coincident power in kW: E Efc

R* & = & (10-84a) 

Thermal output rate in kW (Btu/h): Q Qfc

R* & & = (10-84b) 

Fuel use rate in kW (Btu/h): F FR* & = & (10-84c) 

Water use rate in L/h (gal/h): w

R* V&

w = V& (10-84d) 

Ventilation airflow rate (if from indoors) in kg/h (lbm/h): on

a

R* m& a = m& (10-84e) 

where the energy and mass flow rates are determined as described in Section 9.2 and are 

averaged over the duration of the steady-state test. 

10.4.3 Fuel use, thermal output, ventilation airflow, and water use rates as functions 

of thermal output and ambient temperature for Type III RFCS 

10.4.3.1 Overview of part-load performance evaluation 

The annual rating procedure for a Type III RFCS, described in Section 10.4.5, requires 

part-load factors that relate performance to ambient temperature and thermal part-load ratio 

which is defined as: 

Q 1h

Q

r R

fc

t × = & (10-85) 

For a Type III RFCS, part-load factors expressed as polynomials that are first-order in 

ambient temperature, Ta, and second-order in thermal part-load ratio, rt, are defined for the 

fuel use rate (Zf), power output (Ze), ventilation airflow rate (Za), and water use rate (Zw). 

The coefficients for each polynomial are determined by linear regression using data from 

the six steady-state tests specified in Table 10.4. For each test, the system is operated to 
45

yield the specified thermal part load ratio, rt. The environmental chamber is operated to 

maintain the specified ambient temperature which is TH or TL for outdoor units and TI for 

indoor units. Based on the recorded data, results are calculated for the power, thermal 

output rate, fuel use rate, ventilation airflow rate (for units drawing air from indoors), and 

water use rate as specified in Section 9.2. These quantities along with the ambient 

temperature are averaged over the duration of the steady-state test. Based on these results 

and the rated values, the thermal part-load ratio, and the part-load factors for the power, 

fuel use rate, ventilation airflow rate, and water use rate are determined. The procedure is 

illustrated in Table 10.4. Data from the final six columns are used in the linear regressions 

to determine the coefficients for the polynomial functions. The definition, functional form, 

and application of each part-load factor are presented in the sections following the table. 

Table 10.4. Procedure for determining part load factors for energy and mass flow rates as 

functions of thermal part-load ratio and ambient temperature for Type III RFCS. 

Test conditions Calculated results 

 rt Ta Mass and energy flows1 Part load factors Independent 

variables 

 Efc & Qfc & F& ma & 3 Vw

& Zf Ze Zw Za rt Ta

1.0 TH 

Midpt TH

Min2 TH Eq. Eq. Eq. N.A. Eq. Eq. Eq. Eq. N.A. Note 1 Note 1

1.0 TL (9-2) (9-3) (9-1) (9-15) (10-86) (10-89) (10-92) 

Midpt TL 

Outdoor 

RFCS 

Min2 TL 

1.0 TI 

Midpt TI Note 1 Note 1

Indoor 

RFCS Min TI

Eq. 

(9-2) 

Eq. 

(9-3) 

Eq. 

(9-1)

Eq. 

(9-10)

Eq. 

(9-15)

Eq. 

(10-86)

Eq. 

(10-89)

Eq. 

(10-92)

Eq. 

(10-98) 

Notes: 1. Quantities are averaged over the duration of the steady state test. 

 2. Minimum fuel cell output is defined by the manufacturer. 

 3. Ventilation airflow rate is determined for units that draw air from indoors. 

10.4.3.2 Part-load factors for outdoor Type IIII RFCS 

Part-load fuel use factor. The part-load fuel use factor, Zf, is the fraction of the rated fuel 

use rate at the conditions of thermal part load ratio, rt, and system ambient temperature, Ta: 

( )

R

t a

f F

F r ,T Z &

& ≡ (10-86) 

The functional form of Zf is: 

( ) a 6 a

2

4 t a 5 t

2 Zf rt ,Ta = e1 + e2rt + e3rt + e r T + e r T + e T (10-87) 

where the coefficients are determined by linear regression of the steady state test data. A 

representative function for the part-load fuel use factor is illustrated in Figure 10.5. 
46

TL, 0.2, 0.3

TL, 0.6, 

0.62

TL, 1, 1.08

TH, 0.2, 0.2

TH, 0.6, 

0.56

TH, 1, 1

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Thermal part-load ratio

Part-load fuel use factor

TL

TH

TL, 1, 0.9

TL, 0.6, 0.5

TL, 0.2, 0.2

TH, 1, 1

TH, 0.6, 0.7

TH, 0.2, 0.4

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Thermal part-load ratio

Part-load power output factor

TL

TH

( ) a 6 a

2

4 t a 5 t

2 Zf rt ,Ta = e1 + e2rt + e3rt + e r T + e r T + e T ( ) a 6 a2

4 t a 5 t

2 Ze rt ,Ta = f1 + f2rt + f3rt + f r T + f r T + f T

e1 = 0.2994 e2 = 0.2625 e3 = 0.5781 f1 = -0.03438 f2 = 0.3125 f3 = 0.5469 

e4 = 0.0125 e5 = -0.00938 e6 = -0.00713 f4 = 0.0125 f5 = -0.01563 f6 = 0.008125 

Figure 10.5. Representative function for Figure 10.6. Representative function for 

 fuel use part-load factor power output part-load factor 

The part-load fuel use factor is used in the annual performance rating procedure to 

determine the fuel use, F& , in kW (Btu/h) as a function of the electrical power and ambient 

temperature according to: 

( ) R F& = Zf rt ,Ta × F& (10-88) 

Part-load power output factor. The part-load power output factor Ze is the fraction of the 

rated output power at conditions of thermal part load ratio, rt, and system ambient 

temperature, Ta: 

( )

R

fc t a

e E

E r ,T Z &

& ≡ (10-89) 

The functional form of Ze is: 

( ) a 6 a

2

4 t a 5 t

2 Ze rt ,Ta = f1 + f2rt + f3rt + f r T + f r T + f T (10-90) 

where the coefficients are determined by linear regression of the steady state test data. A 

representative function for the part-load power output factor is illustrated in Figure 10.6. 

The part-load power output factor is used in the annual performance rating procedure to 

determine the power output, Efc & , in kW as a function of the thermal part-load ratio and the 

ambient temperature according to: 

( ) R E& fc = Ze rt ,Ta E& (10-91) 
47

Part-load water use factor. The part-load water use factor Zw is the fraction of the rated 

water use rate required at the conditions of thermal part load ratio, rt, and ambient 

temperature Ta: 

( )

R

w

w t a

w V

V r ,T Z &

& ≡ (10-92) 

The functional form of Zw is: 

( ) a 6 a

2

4 t a 5 t

2 Zw rt ,Ta = g1 + g2rt + g3rt + g r T + g r T + g T (10-93) 

where the coefficients are determined by linear regression of the steady state test data. A 

representative function for the part-load water use factor is illustrated in Figure 10.7. 

The part-load water use factor is used in the annual performance rating procedure to 

determine the water use, Vw

& , in L/h (gal/h) as a function of the thermal part-load ratio and 

the ambient temperature according to: 

( ) R

w w t Ta Vw V Z r , & = & (10-94) 

10.4.3.3 Part-load factors for indoor Type III RFCS 

The part-load factors for an indoor Type III RFCS are defined and applied in the same way 

as those for an outdoor unit except there is no functional dependence on ambient 

temperature because it is assumed to remain constant at a value of TI. Thus the forms for 

the part-load functions are simplified to: 

( ) 2

t

*

t 3

*

2

*

t 1

*

f Z r = e + e r + e r (10-95) 

( ) 2

t

*

t 3

*

2

*

t 1

*

e Z r = f + f r + f r (10-96) 

( ) 2

t

*

t 3

*

2

*

t 1

*

w Z r = g + g r + g r (10-97) 

In addition, if an indoor RFCS draws ventilation air from the residence, it imposes air 

conditioning and heating loads. To model these loads, a part-load ventilation airflow factor 

is defined for units that use indoor air. The part-load ventilation airflow factor, Za, is the 

fraction of the rated ventilation airflow required at the conditions of thermal part-load ratio, 

rt, and system ambient temperature TI: 

( )

R*

a

t

*

* a

a m

m r Z &

& ≡ for Ta = TI (10-98) 

The functional form of Za is: 

( ) 2

t

*

t 3

*

2

*

t 1

*

a Z r = h + h r + h r (10-99) 

where the coefficients are determined by linear regression of the steady state test data. A 

representative function for the part-load ventilation airflow factor is illustrated in Figure 

10.8. 
48

TL, 1, 1.1

TL, 0.6, 

0.65

TL, 0.2, 

0.33

TH, 1, 1

TH, 0.6, 

0.57

TH, 0.2, 

0.23

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Thermal part-load ratio

Part-load water use factor

TL

TH

TI, 1, 1

TI, 0.6, 

0.5975

TI, 0.2, 

0.2645

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Thermal part-load ratio

Part-load ventilation airflow factor

TI

( ) a 6 a

2

4 t a 5 t

2 Zf rt ,Ta = g1 + g2rt + g3rt + g r T + g r T + g T ( ) 2

t

*

t 3

*

2

*

e 1

*

a Z r = h + h r + h r

g1 = 0.3125 g2 = 0.3625 g3 = 0.5000 * h1 = 0.1241 * h2 = 0.6588 *

3 h = 0.2172 

g4 = 0.00750 g5 = -0.00625 g6 = -0.00625 

Figure 10.7. Representative function for Figure 10.8. Representative function for 

 part-load water use factor part-load ventilation airflow 

 factor 

The part-load ventilation airflow factor is used in the annual performance rating procedure 

to determine the ventilation airflow rate, * ma & , in kg/h (lbm/h) as a function of the thermal 

part load ratio: 

( ) R*

t a

*

a

* m& a = Z r m& (10-100) 

10.4.4 Analysis of Simulated Use Data for Type III RFCS 

10.4.4.1 Hot water simulated use data for outdoor Type III RFCS 

For an outdoor Type II RFCS, the thermal energy provided by the RFCS for water heating 

is determined for each hour of a hot water simulated use test performed at ambient 

temperatures of TH and TL. In the rating procedure of Section 10.4.5, the values at a 

particular ambient temperature are estimated based on interpolation between the results at 

TH and those at TL. A similar approach is applied for electrical energy, fuel use, and water 

use. 

Thermal output for water heating. Using data from the hot water simulated use test, hourly 

values in kW⋅h (Btu) for thermal energy provided for water heating by the RFCS at TH and 

TL, respectively, are determined from: 

Q Q ( ) T dt H

t t

t t whH,i wh

i

i 1 ∫

=

= −

= & (10-101a) 
49

Q Q ( ) T dt L

t t

t t whL,i wh

i

i 1 ∫

=

= −

= & (10-101b) 

where: 

Qwh & is the thermal output rate evaluated from Eq. 9-4 at the indicated values of 

power and ambient temperature, kW (Btu/h) 

QwhH,i is the thermal energy provided by the RFCS/preheat tank during hour i at 

ambient temperature TH, kW⋅h (Btu) 

QwhL,i is the thermal energy provided by the RFCS/preheat tank during hour i at 

ambient temperature TL, kW⋅h (Btu) 

The average thermal loss, Q and loss,L Q , are quantities of thermal energy produced by loss,H

the fuel cell unit during the hot water simulated use tests at ambient temperatures TL and 

TH, respectively, that cannot be supplied as useful heat. They are calculated over the hot 

water simulated use test as: 

Q ( ) T dt 1 h 1 Q L

t t

t t loss,L loss

i

i 1

×



 





τ = ∫

=

= −

& (10-102a) 

Q ( ) T dt 1 h 1 Q H

t t

t t loss,H loss

i

i 1

×



 





τ = ∫

=

= −

& (10-102b) 

where: 

Qloss

& is the thermal loss rate evaluated from Eq. 9-5 at the indicated value of ambient 

temperature. 

Electrical energy produced in conjunction with water heating. Using data from the hot 

water simulated use test, hourly values in kW⋅h for the electrical energy provided while 

meeting the loads imposed during the hot water simulated use test at TH and TL, 

respectively, are determined from: 

E E ( ) T dt H

t t

t t whH,i fc

i

i 1 ∫

=

= −

= & (10-103a) 

E E ( ) T dt L

t t

t t whL,i fc

i

i 1 ∫

=

= −

= & (10-103b) 

where: 

Efc & is the power output evaluated from Eq. 9-2 at the indicated ambient 

temperature, kW 

EwhH,i is the electrical energy provided by the RFCS as a result of meeting the water 

heating load during hour i at ambient temperature TH, kW⋅h 

EwhL,i is the electrical energy provided by the RFCS as a result of meeting the water 

heating load during hour i at ambient temperature TL, kW⋅h 
50

Fuel use for water heating. Using data from the hot water simulated use test, hourly values 

in kWh (Btu) for the fuel used to meet the loads imposed during the hot water simulated 

use test at TH and TL, respectively, are determined from: 

F F( ) T dt H

t t

t t whH,i

i

i 1 ∫

=

= −

= & (10-104a) 

F F( ) T dt L

t t

t t whL,i

i

i 1 ∫

=

= −

= & (10-104b) 

where: 

F& is the fuel use rate evaluated from Eq. 9-1 at the indicated ambient 

temperature, kW 

FwhH,i is the fuel used by the RFCS as a result of meeting the water heating load 

during hour i at ambient temperature TH, kW⋅h 

FwhL,i is the fuel used by the RFCS as a result of meeting the water heating load 

during hour i at ambient temperature TL, kW⋅h 

Water use for water heating. Using data from the hot water simulated use test, hourly 

values in L (gal) for the water used by the RFCS to meet the loads imposed during the hot 

water simulated use test at TH and TL, respectively, are determined from: 

( ) Vw,whH,i = Vw,i TH (10-105a) 

( ) Vw,whL,i = Vw,i TL (10-105b) 

where: 

Vw

& is the water use rate evaluated from Eq. 9-16 at the indicated ambient 

temperature, L/h (gal/h) 

Vw,whH,i is the water used by the RFCS as a result of meeting the water heating load 

during hour i at ambient temperature TH, L (gal) 

Vw,whL,i is the water used by the RFCS as a result of meeting the water heating load 

during hour i at ambient temperature TL, L (gal) 

10.4.4.2 Hot water simulated use data for indoor Type III RFCS 

For an indoor Type III RFCS, the thermal energy for water heating, electrical energy, fuel 

use, and water use during hour i are each evaluated from the following expressions using 

data from the hot water simulated use test conducted at ambient temperature TI. 

Thermal output for water heating. The thermal energy provided for water heating during 

hour i expressed in kW⋅h is given by: 

Q Q ( ) T dt I

t t

t t wh

*

wh,i

i

i 1 ∫

=

= −

= & (10-106) 

where ( ) Qwh TI & is the rate at which thermal energy is used for water heating evaluated from 

Eq. 9-4. 
51

The average thermal loss, * Q , in kW loss ⋅h (Btu) is hourly thermal energy produced by the 

fuel cell unit that cannot be supplied as useful heat. It is calculated over the hot water 

simulated use test as: 

Q ( ) T dt 1 h

τ

1 Q I

 t t

 t t loss

*

loss

i

i 1

×



 



 = ∫

=

= −

& (10-107) 

Where Qloss & is the thermal loss rate evaluated from Eq. 9-5 at the indoor ambient temperature, TI. 

Electrical energy produced in conjunction with water heating. The electrical energy 

produced in conjunction with water heating during hour i expressed in kW⋅h is given by: 

E E ( ) T dt I

t t

t t fc

*

wh,i

i

i 1 ∫

=

= −

= & (10-108) 

where, ( ) Efc TI & is the power output evaluated from Eq. 9-2. 

Fuel use for water heating. The fuel used for water heating during hour i expressed in 

kW⋅h is given by: 

F F( ) T dt I

t t

t t

*

wh,i

i

i 1 ∫

=

= −

= & (10-109) 

where ( ) F TI & is the fuel use rate evaluated from Eq. 9-1. 

Water use for water heating. The water used by the RFCS for water heating during hour i 

expressed in L (gal) is given by: 

w,i

* Vw,i = V (10-110) 

where V is the water use rate evaluated from Eq. 9-16. w,i

Ventilation airflow for water heating. For an indoor RFCS that uses indoor air for 

ventilation, the ventilation airflow required by the unit must be determined because it 

affects the heating and cooling loads for the residence. The ventilation airflow during hour 

i expressed in kg/h (lbm/h) is given by: 

a,i

* m& a,wh,i = m (10-111) 

where m is determined from Eq. 9-11 using results from the how water simulated use a,i

test conducted at TI. 

10.4.5 Annual Performance Rating Procedure for Type III RFCS 

10.4.5.1 Overview of rating procedure 

Using the procedure described in the Section 10.4.2, steady-state test results can be used to 

describe the fuel use rate, thermal output rate, ventilation airflow rate, and water use rate of 

a Type III RFCS as functions of the thermal part load ratio and the ambient temperature. 
52

The approach described in Section 10.4.3 can be used to establish the performance of a 

RFCS (in combination with a preheat tank) during each hour of a hot water simulated use 

test. This section describes how the steady-state functions and the hour-by-hour results for 

the hot water simulated use test can be combined to simulate the annual performance of a 

Type III RFCS. 

10.4.5.2 Rating procedure for outdoor Type III RFCS 

For an outdoor Type III RFCS, the thermal energy supplied by the fuel cell system is 

determined by thermal loads of the residence for water heating and space heating. For each 

hour, i, the thermal energy for water heating and the corresponding electrical energy 

output, fuel use, and water use are determined by using the ambient temperature for each 

hour to interpolate between the results from the hot water simulated use tests conducted at 

high (TH) and low (TL) ambient temperatures: 

( )

H L

a,i L

wh,i whL,i whH,i whL,i T T

T T

Q Q Q Q −

− = + − × (10-112) 

( )

H L

a,i L

loss,i loss,L loss,H loss,L T T

T T

Q Q Q Q −

− = + − × (10-113) 

( )

H L

a,i L

wh,i whL,i whH,i whL,i T T

T T

F F F F −

− = + − × (10-114) 

( )

H L

a,i L

wh,i whL,i whH,i whL,i T T

T T

E E E E −

− = + − × (10-115) 

( )

H L

a,i L

w,wh,i w,whL,i w,whH,i w,whL,i T T

T T

V V V V −

− = + − × (10-116) 

The thermal energy available from the RFCS at full load and the temperature during hour I, 

Qav,i, in kW⋅h (Btu) is determined by the thermal output part-load factor and the rated 

output: 

Q Z ( ) T Q 1h R

av,i = q a,i × × & (10-117) 

The thermal energy that is actually used is the minimum of the available thermal energy 

and the thermal energy needed for space heating during hour i, QLs,i, and water heating, 

Qwh,i, each expressed in kW⋅h (Btu): 

[ ] fc,i av,i QLs,i Qwh,i Qloss,i Q = min Q , + + (10-118) 

The thermal energy applied to space heating, Qsh,i, in kW⋅h (Btu) is the difference between 

the total thermal output and the thermal energy required for water heating: 

Qsh,i = Qfc,i − Qwh,i − Qloss,i (10-119) 

There is likely some synergy associated with combining the water heating and space 

heating loads (e.g., less cycling of the system). However, the amounts of electricity 

produced, fuel used, and water used in the process of meeting the combined water heating 
53

and space heating requirements cannot be less than the amounts associated with water 

heating alone as reflected in the simulated use tests. Likewise, the amounts of electricity 

produced, fuel used, and water used in the process of meeting the combined water heating 

and space heating requirements cannot be less than the amounts associated with meeting 

the sum of the water heating and space heating loads while operating in steady state mode. 

Thus, the electrical output, Ei , in kW⋅h, the fuel use, Fi in kW⋅h (Btu), and the water use 

V , in L (gal) are each estimated as the w,i maximum of the respective simulated use and 

steady values: 

[ ] i wh,i Fss,i F = max F , (10-120) 

[ ] fc,i wh,i Efc,ss,i E = max E , (10-121) 

[ ] w,i w,wh,i Vw,ss,i V = max V , (10-122) 

where variables associated with water heating are as previously defined and: 

Efc,ss,i is the electrical energy produced if the RFCS operated in steady state mode 

to meet the combined water heating and space heating requirement, kW⋅h 

F is fuel used if the RFCS operated in ss,i steady state mode to meet the combined 

water heating and space heating requirement, kW⋅h (Btu) 

V is the water used if the RFCS operated in steady state mode to meet the w,ss,i

combined water heating and space heating requirement, L (gal) 

Values for the electricity produced, fuel used, and water used in conjunction with meeting 

only the water heating requirement (Fwh,i, Ewh,i, and Vw,wh,i) are determined by Eqs. 10-114, 

10-115, and 10-116 respectively. The remaining mass and energy flows used in Eqs. 10-

124 to 10-126 (i.e., Efc,ss,i , Fss,i , and Vw,ss,i) are determined based on the thermal part-load 

ratio corresponding to the combined water heating and space heating load: 

Q 1h

Q

r R

fc,i

t,i

× = & (10-123) 

The thermal part-load ratio is used in conjunction with the functions previously defined for 

the part-load factors to determine the values for electricity produced, fuel used, and water 

used in conjunction with meeting both water heating and space heating requirements if 

RFCS operation is steady: 

E Z ( ) r ,T E 1h R

fc,ss,i = e t,i a,i × & × (10-124) 

( ) , 1h , = , , × × R Fss i Z f rt i Ta i F& (10-125) 

( ) , 1h , , = , , × × R

w ss i w t i Ta i Vw V Z r & (10-126) 

The electrical energy that is used by the residence is the minimum of the electrical energy 

available from the fuel cell and the electrical energy requirement of the residence: 
54

[ ] i fc,i EL,i E = min E , (10-127) 

The remaining electrical energy (if any), Gi, in kW⋅h is supplied to the electrical grid: 

Gi = Efc,i − Ei (10-128) 

Annual results are summed over each representative day, weighted by the number of 

occurrences of each representative day, and reported in the format described in Section 

10.1. 

10.4.5.3 Rating procedure for indoor Type III RFCS 

For an indoor Type III RFCS, there is no dependence of performance on ambient 

temperature because the ambient temperature is assumed to remain constant at TI. Thus, 

the thermal energy output as well as the corresponding electrical energy output, fuel use, 

and water use are determined strictly by the water heating and space heating requirements 

of the residence. 

The water heating requirement as well as the corresponding electrical energy output, fuel 

use, water use, and ventilation airflow quantity are determined from the hot water 

simulated use test as described by Eqs. 10-106, 10-108, 10-109, 10-110, and 10-111. 

If the unit uses indoor ventilation air, the thermal energy that is required for heating the 

ventilation air associated with water heating, * ∆Qa,wh,i , in kJ (Btu) is given by: 

Q max[m 1h c (T T );0] pa iah oa

*

a,wh,i

* ∆ a,wh,i = & × × − (10-129) 

where: 

cpa is the constant pressure specific heat for air at assumed conditions of 5 oC 

(40 oF) which is 1.004 kJ/kg⋅oC (0.240 Btu/lbm⋅oF) 

Toa is the temperature of the outdoor air at hour i, oC (oF) 

Tiah is the temperature of the indoor air at heating conditions which is 21 oC (70 oF) 

The thermal energy used for heating water and the associated ventilation air is subtracted 

from the rated RFCS thermal output, * Qfc,i , to determine a first approximation for the 

thermal energy available for space heating from the RFCS:

*

loss,i

*

a,wh,i

*

wh,i

*

fc,i

* Q1av,i = Q − Q − ∆Q − Q (10-130) 

where the correction, * ∆Qa,wh,i , is zero if the unit does not use indoor air for ventilation. A 

first approximation to the thermal output of an indoor Type III RFCS for space heating is 

the minimum of the available thermal energy and the space heating load: 

( ) Ls,i

*

1av,i

* Q1sh,i = min Q ,Q (10-131) 

This first approximation must be corrected to reflect the increase in thermal energy 

required by the residence due to heating ventilation air required by the RFCS as it supplies 
55

space heating. The quantity of air is determined by the first approximation to the thermal 

part-load ratio: 

Q 1h

Q Q Q

r R*

fc

*

loss,i

*

wh,i

*

1sh,i

1t,i

×

+ + = & (10-132) 

The first approximation to the thermal part-load ratio is used in conjuction with the partload ventilation air factor to determine the first approximation to the mass of ventilation air 

associated with space heating, * ma,sh,i & in kg (lbm): 

( ) R*

1t,i a

*

a

* m& a,sh,i = Z r m& (10-133) 

The thermal energy that is required for heating the ventilation air associated with space 

heating is, ∆Qa,sh,i, in kJ (Btu): 

Q max[m 1h c (T T );0] pa iah oa

* ∆ a,sh,i = & a,sh,i × × − (10-134) 

The net thermal energy for space heating supplied by the RFCS is given by: 

[( ) ] Ls,i

*

loss,i

*

a,sh,i

*

a,wh,i

*

wh,i

* R* Qsh,i = min Q& ×1h − Q − ∆Q − ∆Q − Q ,Q (10-135) 

There is likely some synergy associated with combining the water heating and space 

heating loads (e.g., less cycling of the system). However, the amounts of electricity 

produced, fuel used, and water used in the process of meeting the combined water heating 

and space heating requirements cannot be less than the amounts associated with water 

heating alone as reflected in the simulated use tests. Likewise, the amounts of electricity 

produced, fuel used, and water used in the process of meeting the combined water heating 

and space heating requirements cannot be less than the amounts associated with meeting 

the sum of the water heating and space heating loads while operating in steady state mode. 

Thus, the electrical output, * E , in kW i ⋅h, the fuel use, * F in kWh (Btu), and the water use i

* Vw,i , in L (gal) are each estimated as the maximum of the respective simulated use and 

steady values: 

[ ] *

ss,i

*

wh,i

* Fi = max F ,F (10-136) 

[ ] *

ss,i

*

wh,i

* Ei = max E ,E (10-137) 

[ ] *

ss,i

*

wh,i

* Vi = max V ,V (10-138) 

where variables associated with water heating are as previously defined and: 

* Efc,ss,i is the electrical energy produced if the RFCS operated in steady state mode 

to meet the combined water heating and space heating requirement, kW⋅h 

* Fss,i is fuel used if the RFCS operated in steady state mode to meet the combined 

water heating and space heating requirement, kW⋅h (Btu) 

* Vw,ss,i is the water used if the RFCS operated in steady state mode to meet the 

combined water heating and space heating requirement, L (gal) 
56

Values for the electricity produced, fuel used, and water used in conjunction with meeting 

the water heating requirement are determined directly from the simulated use test (Eqs. 10-

108, 10-109, and 10-110). The remaining mass and energy flows for steady state operation 

to meet the combined space heating and water heating requirements ( * Efc,ss,i , * Fss,i , * Vw,ss,i) 

are based on the gross thermal energy output of the RFCS: 

*

loss,i

*

a,sh,i

*

a,wh,i

*

wh,i

*

sh,i

* Qfc,i = Q + Q + ∆Q + ∆Q + Q (10-139) 

This gross thermal output is used to determine a thermal part-load ratio: 

Q 1h

Q

r R*

*

* fc,i

t,i

× = & (10-140) 

For operation in a steady mode to meet both water heating and space heating requirements, 

the electrical energy output, fuel use, and water use are determined by this thermal part 

load ratio: 

E Z (r ) E 1h * R*

t,i

*

e

*

fc,ss,i = × & × (10-141) 

( ) 1h * *

,

* *

, = × × R Fss i Z f rt i F& (10-142) 

( ) 1h * *

,

* *

, , = × × R

w ss i w t i Vw V Z r & (10-143) 

The electrical energy that is used by the residence is the minimum of the electrical energy 

available from the fuel cell and the electrical energy requirement of the residence, EL,i, in 

kW⋅h: 

[ ] L,i

*

fc,i

* Ei = min E ,E (10-144) 

The remaining power (if any), * G , in kW i ⋅h is supplied to the electrical grid: 

*

i

*

fc,i

* Gi = E − E (10-145) 

Annual results are summed over each representative day, weighted by the number of 

occurrences of each representative day, and reported in the format described in Section 

10.1. 
57

10.5 Rated and annual performance for grid interconnected, water heating only Type IV 

RFCS 

10.5.1 Rated performance for outdoor Type IV RFCS 

The RFCS is operated to yield maximum thermal output at an ambient temperature of TH. 

The rated values for the energy and mass flow rates are given by: 

Coincident power in kW: E Efc & R = & (10-146a) 

Thermal output rate in kW (Btu/h): Q Qss & R & = (10-146b) 

Fuel use rate in kW (Btu/h): F F& R = & (10-146c) 

Water use rate in L/h (gal/h): w

R V&

w = V& (10-146d) 

where the energy and mass flow rates are determined as described in Section 9.2 and are 

averaged over the duration of the steady-state test. 

10.5.2 Rated performance for Type IV RFCS 

The RFCS is operated to yield maximum thermal output at an ambient temperature of TI

for indoor units. The rated values for the energy and mass flow rates are given by: 

Coincident power in kW: E Efc & R = & (10-147a) 

Thermal output rate in kW (Btu/h): Q Qss & R & = (10-147b) 

Fuel use rate in kW (Btu/h): F F& R = & (10-147c) 

Water use rate in L/h (gal/h): w

R V&

w = V& (10-147d) 

Ventilation airflow rate (if indoor air) in kg/h (lbm/h): on

a

R m& a = m& (10-147e) 

where the energy and mass flow rates are determined as described in Section 9.2 and are 

averaged over the duration of the steady-state test. 

10.5.3 Analysis of Simulated Use Data for Type IV RFCS 

10.5.3.1 Hot water simulated use data for outdoor Type IV RFCS 

For an outdoor Type IV RFCS, the thermal energy provided by the RFCS for water heating 

is determined for each hour of a hot water simulated use test performed at ambient 

temperatures of TH and TL. In the rating procedure of Section 10.5.4, the values at a 

particular ambient temperature are estimated based on interpolation between the results at 

TH and those at TL. A similar approach is applied for electrical energy, fuel use, and water 

use. 

Thermal output for water heating. Using data from the hot water simulated use test, hourly 

values in kW⋅h (Btu) for thermal energy provided for water heating by the RFCS at TH and 

TL, respectively, are determined from: 

Q Q ( ) T dt H

t t

t t whH,i wh

i

i 1 ∫

=

= −

= & (10-148a) 
58

Q Q ( ) T dt L

t t

t t whL,i wh

i

i 1 ∫

=

= −

= & (10-148b) 

where: 

Qwh & is the thermal output rate evaluated from Eq. 9-4 at the indicated values of 

power and ambient temperature, kW (Btu/h) 

QwhH,i is the thermal energy provided by the RFCS/preheat tank during hour i at 

ambient temperature TH, kWh (Btu) 

QwhL,i is the thermal energy provided by the RFCS/preheat tank during hour i at 

ambient temperature TL, kW⋅h (Btu) 

Electrical energy produced in conjunction with water heating. Using data from the hot 

water simulated use test, hourly values in kW⋅h for the electrical energy provided while 

meeting the loads imposed during the hot water simulated use test at TH and TL, 

respectively, are determined from: 

E E ( ) T dt H

t t

t t whH,i fc

i

i 1 ∫

=

= −

= & (10-149a) 

E E ( ) T dt L

t t

t t whL,i fc

i

i 1 ∫

=

= −

= & (10-149b) 

where: 

Efc & is the power output evaluated from Eq. 9-2 at the indicated ambient 

temperature, kW 

EwhH,i is the electrical energy provided by the RFCS as a result of meeting the water 

heating load during hour i at ambient temperature TH, kW⋅h 

EwhL,i is the electrical energy provided by the RFCS as a result of meeting the water 

heating load during hour i at ambient temperature TL, kW⋅h 

Fuel use for water heating. Using data from the hot water simulated use test, hourly values 

in kW⋅h (Btu) for the fuel used to meet the loads imposed during the hot water simulated 

use test at TH and TL, respectively, are determined from: 

F F( ) T dt H

t t

t t whH,i

i

i 1 ∫

=

= −

= & (10-150a) 

F F( ) T dt L

t t

t t whL,i

i

i 1 ∫

=

= −

= & (10-150b) 

where: 

F& is the fuel use rate evaluated from Eq. 9-1 at the indicated ambient 

temperature, kW 

FwhH,i is the fuel used by the RFCS as a result of meeting the water heating load 

during hour i at ambient temperature TH, kW⋅h 
59

FwhL,i is the fuel used by the RFCS as a result of meeting the water heating load 

during hour i at ambient temperature TL, kW⋅h 

Water use for water heating. Using data from the hot water simulated use test, hourly 

values in L (gal) for the water used by the RFCS to meet the loads imposed during the hot 

water simulated use test at TH and TL, respectively, are determined from: 

( ) Vw,whH,i = Vw,i TH (10-151a) 

( ) Vw,whL,i = Vw,i TL (10-151b) 

where: 

V is the water use rate evaluated from Eq. 9-16 at the indicated ambient w,i

temperature, L/h (gal/h) 

Vw,whH,i is the water used by the RFCS as a result of meeting the water heating load 

during hour i at ambient temperature TH, L (gal) 

Vw,whL,i is the water used by the RFCS as a result of meeting the water heating load 

during hour i at ambient temperature TL, L (gal) 

10.5.3.2 Hot water simulated use data for indoor Type IV RFCS 

For an indoor Type IV RFCS, the thermal energy for water heating, electrical energy, fuel 

use, and water use during hour i are each evaluated from the following expressions using 

data from the hot water simulated use test conducted at ambient temperature TI. 

Thermal output for water heating. The thermal energy provided for water heating during 

hour i expressed in kW⋅h is given by: 

Q Q ( ) T dt I

t t

t t wh

*

wh,i

i

i 1 ∫

=

= −

= & (10-152) 

where ( ) Qwh TI & is the rate at which thermal energy is used for water heating evaluated from 

Eq. 9-4. 

Electrical energy produced in conjunction with water heating. The electrical energy 

produced in conjunction with water heating during hour i expressed in kW⋅h is given by: 

E E ( ) T dt I

t t

t t fc

*

wh,i

i

i 1 ∫

=

= −

= & (10-153) 

where, ( ) Efc TI & is the power output evaluated from Eq. 9-2. 

Fuel use for water heating. The fuel used for water heating during hour i expressed in 

kW⋅h is given by: 

F F( ) T dt I

t t

t t

*

wh,i

i

i 1 ∫

=

= −

= & (10-154) 

where ( ) F TI & is the fuel use rate evaluated from Eq. 9-1. 
60

Water use for water heating. The water used by the RFCS for water heating during hour i 

expressed in L (gal) is given by: 

w,i

* Vw,i = V (10-155) 

where V is the water use rate evaluated from w,i Eq. 9-16 using results from the how water 

simulated use test conducted at TI. 

Ventilation airflow for water heating. For an indoor RFCS that draws ventilation air from 

the indoors, the ventilation airflow required by the unit must be determined because it 

affects the heating and cooling loads for the residence. The ventilation airflow during hour 

i expressed in kg/h (lbm/h) is given by: 

a,i

* ma,wh,i = m (10-156) 

where m is determined from Eq. 9-11 using results from the how water simulated use a,i

test conducted at TI. 

10.5.4 Annual Performance Rating Procedure for Type IV RFCS 

10.5.4.1 Overview of rating procedure 

The approach described in Section 10.5.3 can be used to establish the performance of a 

RFCS (in combination with a preheat tank) during each hour of a hot water simulated use 

test. This section describes how the hour-by-hour results for the hot water simulated use 

test can be used to simulate the annual performance of a Type IV RFCS. 

10.5.4.2 Rating procedure for outdoor Type IV RFCS 

For an outdoor Type IV RFCS, the thermal energy supplied by the fuel cell system is 

determined by thermal loads of the residence for water heating. For each hour, i, the 

thermal energy for water heating and the corresponding electrical energy output, fuel use, 

and water use are determined by using the ambient temperature for each hour to interpolate 

between the results from the hot water simulated use tests conducted at high (TH) and low 

(TL) ambient temperatures: 

( )

H L

a,i L

wh,i whL,i whH,i whL,i T T

T T

Q Q Q Q −

− = + − × (10-157) 

( )

H L

a,i L

i whL,i whH,i whL,i T T

T T

F F F F −

− = + − × (10-158) 

( )

H L

a,i L

fc,i whL,i whH,i whL,i T T

T T

E E E E −

− = + − × (10-159) 

( )

H L

a,i L

w,i w,whL,i w,whH,i w,whL,i T T

T T

V V V V −

− = + − × (10-160) 
61

The electrical energy that is used by the residence is the minimum of the electrical energy 

available from the fuel cell and the electrical energy requirement of the residence: 

[ ] i fc,i EL,i E = min E , (10-161) 

The remaining electrical energy (if any), Gi, in kW⋅h is supplied to the electrical grid: 

Gi = Efc,i − Ei (10-162) 

Annual results are summed over each representative day, weighted by the number of 

occurrences of each representative day, and reported in the format described in Section 

10.1. 

10.5.4.3 Rating procedure for Type IV RFCS 

For an indoor Type IV RFCS, there is no dependence of performance on ambient 

temperature because the ambient temperature is assumed to remain constant at TI. Thus, 

the thermal energy output as well as the corresponding electrical energy output, fuel use, 

and water use are determined strictly by the water heating requirements of the residence. 

The water heating requirement as well as the corresponding electrical energy output, fuel 

use, water use, and ventilation airflow quantity are determined from the hot water 

simulated use test as described by Eqs. 10-152 to 10-156. 

If the unit draws ventilation air from the indoors, the thermal energy that is required for 

heating the ventilation air associated with water heating is, * ∆Qa,wh,i , in kJ (Btu): 

Q max[m 1h c (T T );0] pa iah oa

*

a,wh,i

* ∆ a,wh,i = & × × − (10-163) 

where: 

cpa is the constant pressure specific heat for air at assumed conditions of 5 oC 

(40 oF), 

Toa is the temperature of the outdoor air at hour i, oC (oF) 

Tiah is the temperature of the indoor air at heating conditions which is 21 oC (70 oF) 

The contribution of an indoor Type IV RFCS (which uses indoor air for ventilation) to the 

energy used by the residence for space heating is simply: 

*

a,wh,i

* Qsh,i = −∆Q (10-164) 

which will be negative, indicating that somewhat more heating is required due to the 

installation of the indoor Type IV RFCS. The contribution will be zero if the unit does not 

draw ventilation air from the indoors. 

The electrical energy that is used by the residence is the minimum of the electrical energy 

available from the fuel cell and the electrical energy requirement of the residence, EL,i, in 

kW⋅h: 

[ ] L,i

*

fc,i

* Ei = min E ,E (10-165) 
62

The remaining power (if any), Gi, in kW⋅h is supplied to the electrical grid: 

*

i

*

fc,i

* Gi = E − E (10-166) 

Annual results are summed over each representative day, weighted by the number of 

occurrences of each representative day, and reported in the format described in Section 

10.1. 
63

11. References 

[1] CSA America, ANSI/CSA America FC 1-2004, Stationary Fuel Cell Power Systems, CSA 

America, 2004 

[2] National Fire Protection Association, NFPA 853, Installation of Stationary Fuel Cell Power 

Plants, 2003 Edition, NFPA, 2003 

[3] American Society of Mechanical Engineers, ASME PTC 50-2002, Fuel Cell Power Systems 

Performance, ASME, New York, 2002. 

[4] M.W. Davis, “Proposed Testing Methodology and Laboratory Facilities for Evaluating 

Residential Fuel Cell Systems,” NISTIR 6848, Jan. 2002 

[5] American Society of Mechanical Engineers, ASME PTC 19.1-1988, Test Uncertainty, 

ASME, New York, 1988. 

[6] U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, “Energy 

Conservation Program for Consumer Products: Test Procedure for Water Heaters”, Federal 

Register, vol.63, No. 90, May 11, 1998. 
A-1

Appendices 

A. Residential Loads and Climate Data for Department of Energy (DOE) Climate 

Regions 

Simulation of the annual energy use of residential fuel cell systems requires knowledge of 

electrical and thermal loads for residences as well as coincident outdoor conditions. This rating 

procedure uses building energy simulation results for a prototypical house located in a 

representative city within each DOE climate zone. The results are aggregated to yield three days 

representing winter, spring/fall, and summer for each of the climate zones. This data is presented 

in the following tables: 

− Table A.1: Climate Data, Thermal Loads, and Electrical Loads for Region 1 (based on 

Jacksonville, FL) 

− Table A.2: Climate Data, Thermal Loads, and Electrical Loads for Region 2 (based on 

Charleston, SC) 

− Table A.3: Climate Data, Thermal Loads, and Electrical Loads for Region 3 (based on 

Memphis, TN) 

− Table A.4: Climate Data, Thermal Loads, and Electrical Loads for Region 4 (based on 

Pittsburgh, PA) 

− Table A.5: Climate Data, Thermal Loads, and Electrical Loads for Region 5 (based on 

Minneapolis, MN) 

− Table A.6: Climate Data, Thermal Loads, and Electrical Loads for Region 6 (based on 

Astoria, OR) 
A-2

Table A.1: Climate Data, Thermal Loads, and Electrical Loads for Region 1 (based on Jacksonville, FL) 

Hr

C g/kg kJ/kg kW kW 

Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Su

1 9.41 17.89 23.75 6.62 11.62 16.26 26.17 47.47 65.26 0.939 0.079 0.000 0.200 0.247 0.42 9.18 17.53 23.45 6.57 11.53 16.14 25.75 46.87 64.62 1.098 0.101 0.000 0.200 0.225 0.33 8.85 17.14 23.14 6.54 11.41 16.01 25.35 46.21 63.92 1.258 0.121 0.000 0.200 0.213 0.34 8.63 16.84 22.92 6.50 11.30 15.96 25.02 45.62 63.59 1.432 0.149 0.000 0.200 0.207 0.25 8.34 16.57 22.71 6.45 11.16 15.81 24.68 44.94 63.04 1.561 0.176 0.000 0.300 0.304 0.36 8.13 16.35 22.58 6.41 11.04 15.66 24.35 44.40 62.51 1.533 0.182 0.000 0.700 0.709 0.87 7.90 16.42 23.03 6.36 11.16 16.01 24.00 44.80 63.93 1.341 0.160 0.000 1.400 1.449 1.68 8.09 17.64 24.77 6.36 11.59 16.74 24.18 47.14 67.49 1.441 0.158 0.000 0.700 0.747 1.09 9.44 19.78 26.65 6.61 12.06 17.22 26.14 50.48 70.74 1.387 0.146 0.000 0.300 0.372 0.6

10 11.52 21.82 28.29 7.02 12.18 17.13 29.25 52.91 72.16 1.088 0.090 0.000 0.200 0.340 0.711 13.60 23.46 29.49 7.28 12.12 17.04 32.02 54.41 73.17 0.781 0.051 0.000 0.200 0.435 0.912 15.15 24.52 30.41 7.37 12.02 16.96 33.81 55.32 73.96 0.595 0.031 0.000 0.200 0.515 1.013 16.01 25.33 31.01 7.33 11.75 16.65 34.68 55.43 73.75 0.453 0.023 0.000 0.202 0.608 1.114 16.54 25.70 31.07 7.32 11.64 16.66 35.16 55.49 73.87 0.374 0.017 0.000 0.206 0.700 1.215 16.71 25.70 30.73 7.31 11.67 16.75 35.33 55.65 73.70 0.356 0.011 0.000 0.209 0.753 1.316 16.63 25.55 30.10 7.27 11.67 16.56 35.09 55.43 72.57 0.322 0.011 0.000 0.309 0.851 1.417 16.01 24.85 29.22 7.34 11.84 16.54 34.70 55.15 71.67 0.279 0.007 0.000 0.713 1.255 1.818 14.52 23.52 28.33 7.38 11.98 16.62 33.28 54.16 70.92 0.195 0.005 0.000 3.537 4.270 4.919 12.82 21.90 27.33 7.22 11.96 16.65 31.13 52.43 69.94 0.223 0.005 0.000 1.612 2.165 2.720 11.60 20.55 26.22 7.04 11.88 16.52 29.46 50.84 68.44 0.202 0.008 0.000 2.516 3.029 3.521 10.80 19.74 25.35 6.93 11.80 16.48 28.35 49.79 67.52 0.187 0.004 0.000 4.706 5.183 5.622 10.30 19.08 24.87 6.87 11.78 16.62 27.68 49.10 67.39 0.283 0.008 0.000 1.704 2.086 2.523 9.98 18.62 24.55 6.80 11.76 16.68 27.16 48.58 67.14 0.462 0.026 0.000 0.901 1.114 1.424 9.69 18.23 24.15 6.72 11.66 16.49 26.70 47.88 66.21 0.737 0.061 0.000 0.300 0.390 0.6

Outdoor Temperature Outdoor Humidity Ratio Outdoor Air Enthalpy Heating Load Electrical Load (no-he

Toa, °C ωoa, g/kg hoa, kJ/kg QLs, kW EL, kW 
A-3

Table A.2: Climate Data, Thermal Loads, and Electrical Loads for Region 2 (based on Charleston, SC) 

Hr

C g/kg kJ/kg kW kW 

Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Su

1 7.14 15.26 22.68 5.42 9.95 15.09 20.83 40.54 61.18 1.934 0.249 0.000 0.201 0.224 0.32 6.87 14.93 22.42 5.38 9.82 14.97 20.45 39.89 60.60 2.175 0.308 0.001 0.201 0.213 0.33 6.59 14.67 22.18 5.31 9.71 14.84 20.01 39.34 60.03 2.389 0.369 0.000 0.201 0.207 0.24 6.32 14.38 21.90 5.25 9.60 14.75 19.58 38.75 59.49 2.612 0.435 0.000 0.201 0.204 0.25 6.02 14.09 21.68 5.19 9.48 14.66 19.12 38.16 59.05 2.727 0.492 0.000 0.309 0.311 0.36 5.78 13.87 21.64 5.15 9.39 14.62 18.76 37.72 58.89 2.642 0.500 0.000 0.708 0.712 0.77 5.53 14.14 22.30 5.11 9.49 14.97 18.41 38.24 60.47 2.398 0.425 0.000 1.380 1.409 1.68 5.77 15.59 23.84 5.16 9.87 15.46 18.80 40.67 63.33 2.508 0.395 0.000 0.708 0.734 0.99 7.09 17.75 25.57 5.39 10.18 15.56 20.69 43.67 65.35 2.416 0.332 0.000 0.309 0.354 0.6

10 8.90 19.78 27.04 5.56 10.21 15.38 22.98 45.81 66.41 1.977 0.203 0.000 0.237 0.326 0.711 10.65 21.31 28.21 5.64 10.10 15.30 24.95 47.12 67.42 1.483 0.110 0.000 0.237 0.373 0.812 12.06 22.41 29.00 5.65 10.05 15.23 26.40 48.11 68.05 1.076 0.063 0.000 0.237 0.424 0.913 13.22 23.06 29.49 5.63 9.98 15.13 27.52 48.59 68.32 0.863 0.046 0.000 0.237 0.475 1.014 13.87 23.40 29.64 5.58 9.96 15.07 28.06 48.89 68.32 0.714 0.037 0.000 0.237 0.519 1.115 14.01 23.39 29.43 5.55 10.00 14.96 28.13 48.99 67.82 0.657 0.036 0.000 0.237 0.544 1.216 13.87 23.01 29.00 5.56 10.09 14.97 28.01 48.80 67.39 0.659 0.049 0.000 0.309 0.620 1.317 13.12 22.17 28.35 5.59 10.13 15.04 27.31 48.07 66.91 0.655 0.061 0.000 0.713 1.026 1.718 11.84 20.77 27.55 5.65 10.18 15.21 26.17 46.75 66.52 0.495 0.035 0.000 3.512 3.979 4.819 10.57 19.23 26.50 5.67 10.22 15.41 24.93 45.29 65.93 0.577 0.040 0.000 1.566 1.897 2.620 9.63 18.03 25.29 5.63 10.25 15.56 23.88 44.12 65.05 0.570 0.048 0.000 2.480 2.793 3.421 8.90 17.20 24.41 5.60 10.27 15.58 23.08 43.32 64.20 0.605 0.049 0.000 4.721 5.007 5.622 8.33 16.60 23.87 5.58 10.23 15.54 22.44 42.62 63.55 0.807 0.063 0.000 1.725 1.951 2.523 7.93 16.12 23.39 5.56 10.16 15.36 21.98 41.95 62.59 1.178 0.112 0.000 0.946 1.059 1.424 7.61 15.67 23.03 5.54 10.06 15.19 21.62 41.25 61.77 1.638 0.188 0.000 0.273 0.315 0.5

Outdoor Temperature Outdoor Humidity Ratio Outdoor Air Enthalpy Heating Load Electrical Load (no-he

Toa, °C ωoa, g/kg hoa, kJ/kg QLs, kW EL, kW 
A-4

Table A.3: Climate Data, Thermal Loads, and Electrical Loads for Region 3 (based on Memphis, TN) 

Hr

C g/kg kJ/kg kW kW 

Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Su

1 4.57 14.43 23.68 3.88 8.63 14.69 14.31 36.35 61.13 3.006 0.436 0.000 0.200 0.229 0.42 4.31 14.06 23.27 3.83 8.57 14.60 13.87 35.83 60.52 3.210 0.530 0.000 0.200 0.218 0.33 3.95 13.69 22.91 3.80 8.52 14.54 13.54 35.31 60.03 3.416 0.621 0.000 0.200 0.210 0.34 3.75 13.32 22.60 3.80 8.49 14.48 13.28 34.89 59.53 3.560 0.698 0.000 0.200 0.206 0.25 3.43 13.04 22.33 3.80 8.47 14.40 13.03 34.54 59.10 3.620 0.761 0.000 0.300 0.304 0.36 3.29 12.87 22.37 3.79 8.48 14.43 12.80 34.39 59.21 3.509 0.762 0.000 0.700 0.711 0.87 3.30 13.38 23.43 3.83 8.66 14.77 13.03 35.37 61.19 3.163 0.623 0.000 1.400 1.444 1.78 3.85 14.83 25.18 3.91 8.96 15.28 13.70 37.58 64.24 3.185 0.587 0.000 0.700 0.748 1.19 4.58 16.58 26.75 4.01 9.20 15.58 14.65 40.00 66.66 3.085 0.542 0.000 0.300 0.362 0.7

10 5.64 18.15 28.12 4.14 9.31 15.64 16.01 41.87 68.15 2.639 0.372 0.000 0.200 0.292 0.811 6.76 19.39 29.12 4.27 9.33 15.64 17.53 43.18 69.19 2.193 0.236 0.000 0.200 0.330 0.912 7.93 20.50 29.89 4.37 9.31 15.53 18.92 44.23 69.74 1.843 0.160 0.000 0.200 0.366 1.013 8.60 21.26 30.32 4.42 9.24 15.50 19.79 44.87 70.13 1.527 0.113 0.000 0.200 0.405 1.114 8.98 21.71 30.58 4.44 9.15 15.27 20.23 45.13 69.84 1.346 0.083 0.000 0.200 0.437 1.215 9.31 21.96 30.78 4.46 9.07 14.94 20.61 45.16 69.15 1.340 0.079 0.000 0.200 0.456 1.316 9.16 21.80 30.60 4.49 8.99 14.78 20.51 44.77 68.55 1.395 0.093 0.000 0.300 0.554 1.417 8.53 21.03 30.12 4.49 8.98 14.67 19.87 44.01 67.79 1.437 0.094 0.000 0.700 0.954 1.818 7.85 20.02 29.47 4.48 8.96 14.65 19.17 42.92 67.06 1.073 0.052 0.000 3.500 3.888 4.919 7.27 18.83 28.51 4.45 8.96 14.73 18.51 41.69 66.23 1.312 0.069 0.000 1.600 1.887 2.720 6.71 17.66 27.19 4.39 8.98 14.89 17.80 40.56 65.30 1.315 0.081 0.000 2.500 2.779 3.521 6.15 16.71 26.13 4.30 8.94 14.96 17.02 39.50 64.35 1.356 0.090 0.000 4.700 4.959 5.622 5.78 15.98 25.27 4.21 8.85 14.91 16.39 38.53 63.41 1.622 0.147 0.000 1.700 1.905 2.523 5.38 15.45 24.59 4.14 8.77 14.85 15.88 37.71 62.52 2.144 0.240 0.000 0.900 1.013 1.424 4.99 14.90 23.96 4.06 8.68 14.75 15.29 36.95 61.68 2.714 0.368 0.000 0.300 0.351 0.6

Outdoor Temperature Outdoor Humidity Ratio Outdoor Air Enthalpy Heating Load Electrical Load (no-he

Toa, °C ωoa, g/kg hoa, kJ/kg QLs, kW EL, kW 
A-5

Table A.4: Climate Data, Thermal Loads, and Electrical Loads for Region 4 (based on Pittsburgh, PA) 

Hr

C g/kg kJ/kg kW kW 

Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall S

1 -2.50 8.71 18.17 2.52 5.88 10.96 3.79 23.57 46.04 5.031 1.492 0.004 0.201 0.202 02 -2.69 8.31 17.69 2.49 5.85 10.92 3.52 23.11 45.46 5.247 1.647 0.006 0.201 0.201 03 -2.90 7.94 17.16 2.47 5.84 10.84 3.27 22.69 44.72 5.453 1.797 0.009 0.201 0.201 04 -3.09 7.53 16.67 2.45 5.79 10.75 3.04 22.16 43.97 5.569 1.927 0.012 0.201 0.201 05 -3.26 7.20 16.36 2.44 5.74 10.70 2.83 21.69 43.54 5.524 2.028 0.017 0.309 0.309 06 -3.43 6.99 16.39 2.43 5.71 10.74 2.63 21.42 43.66 5.279 1.986 0.015 0.708 0.708 07 -3.54 7.12 17.28 2.44 5.77 11.06 2.54 21.70 45.40 4.940 1.751 0.008 1.380 1.382 18 -3.60 8.06 19.00 2.46 5.93 11.46 2.52 23.05 48.18 5.144 1.776 0.008 0.708 0.710 09 -3.22 9.48 20.79 2.52 6.09 11.68 3.07 24.90 50.57 5.222 1.735 0.009 0.309 0.312 0

10 -2.42 10.88 22.41 2.61 6.20 11.71 4.10 26.61 52.31 4.976 1.480 0.005 0.237 0.244 011 -1.48 12.16 23.79 2.70 6.25 11.61 5.27 28.03 53.48 4.588 1.229 0.003 0.237 0.253 012 -0.63 13.26 24.87 2.77 6.25 11.55 6.30 29.15 54.42 4.258 1.035 0.001 0.237 0.263 013 0.07 14.19 25.69 2.82 6.24 11.43 7.15 30.08 54.95 3.980 0.926 0.000 0.237 0.278 014 0.57 14.78 26.13 2.85 6.25 11.32 7.71 30.68 55.12 3.873 0.861 0.000 0.237 0.292 015 0.96 15.02 26.34 2.88 6.20 11.23 8.18 30.82 55.10 3.844 0.846 0.000 0.237 0.299 016 1.11 15.07 26.33 2.89 6.11 11.17 8.37 30.62 54.94 3.794 0.867 0.000 0.309 0.372 017 0.79 14.74 25.89 2.85 6.05 11.18 7.95 30.13 54.51 3.766 0.848 0.000 0.713 0.777 118 0.23 14.03 25.20 2.81 6.04 11.17 7.27 29.39 53.77 3.030 0.622 0.000 3.509 3.629 419 -0.31 13.06 24.24 2.77 6.00 11.21 6.62 28.30 52.92 3.408 0.761 0.000 1.566 1.649 220 -0.75 12.06 22.89 2.74 5.93 11.25 6.10 27.11 51.63 3.295 0.707 0.000 2.480 2.558 321 -1.17 11.16 21.59 2.73 5.91 11.21 5.66 26.15 50.19 3.281 0.674 0.000 4.721 4.789 522 -1.53 10.32 20.50 2.70 5.94 11.15 5.22 25.36 48.93 3.637 0.793 0.000 1.725 1.773 223 -1.88 9.65 19.53 2.64 5.93 11.08 4.73 24.66 47.75 4.195 1.034 0.000 0.946 0.963 124 -2.21 9.14 18.74 2.59 5.89 11.03 4.27 24.05 46.82 4.808 1.328 0.002 0.273 0.277 0

Outdoor Temperature Outdoor Humidity Ratio Outdoor Air Enthalpy Heating Load Electrical Load (no-h

Toa, °C ωoa, g/kg hoa, kJ/kg QLs, kW EL, kW 
A-6

Table A.5: Climate Data, Thermal Loads, and Electrical Loads for Region 5 (based on Minneapolis, MN) 

Hr

C g/kg kJ/kg kW kW 

Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Su

1 -10.15 6.31 18.44 1.55 5.04 11.17 -6.33 19.05 46.88 6.575 2.109 0.000 0.201 0.202 0.22 -10.44 5.96 17.97 1.53 5.02 11.02 -6.67 18.63 45.99 6.664 2.279 0.000 0.201 0.201 0.23 -10.69 5.67 17.48 1.52 5.00 10.85 -6.96 18.28 45.08 6.761 2.445 0.000 0.201 0.201 0.24 -10.90 5.41 17.23 1.50 4.99 10.79 -7.21 18.00 44.67 6.822 2.572 0.001 0.201 0.201 0.25 -11.06 5.13 17.19 1.49 4.97 10.79 -7.40 17.66 44.62 6.899 2.666 0.002 0.309 0.309 0.36 -11.24 4.92 17.26 1.47 4.96 10.88 -7.63 17.44 44.91 6.840 2.600 0.001 0.708 0.708 0.77 -11.40 5.44 18.13 1.45 5.06 11.08 -7.84 18.21 46.33 6.578 2.282 0.000 1.380 1.383 1.48 -11.48 6.54 19.59 1.44 5.19 11.30 -7.95 19.67 48.38 6.794 2.260 0.000 0.708 0.710 0.79 -11.21 7.75 21.05 1.45 5.32 11.46 -7.66 21.20 50.27 6.713 2.245 0.000 0.309 0.312 0.3

10 -10.45 8.95 22.30 1.48 5.43 11.59 -6.82 22.69 51.89 6.337 1.967 0.000 0.237 0.242 0.311 -9.38 10.08 23.36 1.54 5.50 11.69 -5.58 24.03 53.24 5.865 1.714 0.000 0.237 0.246 0.412 -8.33 11.15 24.28 1.61 5.54 11.75 -4.35 25.23 54.33 5.575 1.538 0.000 0.237 0.254 0.413 -7.46 11.87 24.91 1.68 5.54 11.77 -3.32 25.96 55.01 5.343 1.401 0.000 0.237 0.261 0.514 -6.84 12.32 25.27 1.72 5.52 11.77 -2.57 26.35 55.39 5.245 1.321 0.000 0.237 0.271 0.615 -6.58 12.71 25.53 1.75 5.50 11.76 -2.25 26.70 55.64 5.251 1.326 0.000 0.237 0.277 0.616 -6.67 12.58 25.50 1.75 5.48 11.75 -2.34 26.52 55.58 5.375 1.369 0.000 0.309 0.348 0.717 -7.13 12.00 25.24 1.73 5.45 11.73 -2.85 25.86 55.26 5.495 1.363 0.000 0.713 0.750 1.118 -7.76 11.30 24.87 1.70 5.41 11.71 -3.56 25.03 54.82 4.713 1.022 0.000 3.509 3.581 4.219 -8.31 10.38 23.92 1.68 5.37 11.71 -4.17 24.01 53.85 5.121 1.153 0.000 1.566 1.614 2.120 -8.70 9.39 22.62 1.66 5.34 11.67 -4.60 22.91 52.43 4.997 1.094 0.000 2.480 2.524 3.021 -9.01 8.49 21.40 1.64 5.28 11.54 -4.97 21.85 50.83 4.995 1.076 0.000 4.721 4.758 5.222 -9.29 7.79 20.45 1.62 5.22 11.41 -5.31 20.99 49.53 5.329 1.239 0.000 1.725 1.751 2.123 -9.61 7.25 19.69 1.59 5.17 11.31 -5.68 20.31 48.51 5.915 1.543 0.000 0.946 0.955 1.124 -9.92 6.76 18.97 1.57 5.12 11.22 -6.05 19.69 47.54 6.472 1.908 0.000 0.273 0.275 0.3

Outdoor Temperature Outdoor Humidity Ratio Outdoor Air Enthalpy Heating Load Electrical Load (no-he

Toa, °C ωoa, g/kg hoa, kJ/kg QLs, kW EL, kW 
A-7

Table A.6: Climate Data, Thermal Loads, and Electrical Loads for Region 6 (based on Astoria, OR) 

Hr

C g/kg kJ/kg kW kW 

Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Sum Wtr Spr/Fall Su

1 4.81 8.12 13.15 4.90 5.96 7.85 17.23 23.21 33.04 2.997 1.624 0.100 0.200 0.200 0.22 4.90 7.95 12.99 4.89 5.91 7.81 17.18 22.88 32.81 3.091 1.763 0.141 0.200 0.200 0.23 4.88 7.81 12.81 4.88 5.89 7.76 17.16 22.69 32.53 3.218 1.907 0.200 0.200 0.200 0.24 4.91 7.74 12.68 4.86 5.87 7.72 17.13 22.55 32.26 3.379 2.047 0.276 0.200 0.200 0.25 4.87 7.68 12.85 4.84 5.87 7.77 17.04 22.53 32.57 3.358 2.090 0.303 0.300 0.300 0.36 4.79 7.80 13.24 4.83 5.88 7.90 16.93 22.61 33.27 3.172 1.933 0.216 0.700 0.700 0.77 4.70 8.02 13.77 4.81 5.92 8.05 16.82 22.93 34.14 2.844 1.563 0.100 1.400 1.400 1.48 4.99 8.83 14.45 4.86 6.12 8.28 17.17 24.26 35.46 2.998 1.575 0.090 0.700 0.700 0.79 5.43 10.01 15.36 4.94 6.36 8.50 17.85 26.08 36.97 3.085 1.620 0.145 0.300 0.300 0.3

10 5.93 11.16 16.32 5.03 6.53 8.69 18.53 27.63 38.41 2.938 1.469 0.102 0.200 0.200 0.211 6.45 11.95 17.11 5.13 6.60 8.83 19.35 28.64 39.52 2.555 1.212 0.048 0.200 0.200 0.212 7.08 12.54 17.62 5.23 6.62 8.95 20.24 29.33 40.42 2.329 1.086 0.021 0.200 0.200 0.213 7.68 13.11 18.25 5.32 6.68 9.06 21.12 30.05 41.26 2.278 1.073 0.022 0.200 0.202 0.214 7.92 13.25 18.44 5.35 6.68 9.06 21.39 30.23 41.53 2.080 0.957 0.016 0.200 0.205 0.215 7.82 13.05 18.29 5.33 6.66 8.99 21.25 29.98 41.20 2.015 0.946 0.013 0.200 0.207 0.216 7.73 12.85 17.99 5.30 6.65 8.88 21.10 29.70 40.63 2.151 1.007 0.015 0.300 0.307 0.317 7.22 12.28 17.43 5.24 6.61 8.73 20.49 29.05 39.67 2.092 0.935 0.006 0.700 0.707 0.718 6.62 11.58 16.67 5.14 6.52 8.56 19.56 28.09 38.44 1.432 0.571 0.002 3.500 3.515 3.519 5.93 10.75 15.80 5.04 6.42 8.40 18.61 26.98 37.18 1.722 0.716 0.006 1.600 1.607 1.620 5.55 10.08 15.16 4.98 6.33 8.29 18.10 26.08 36.15 1.564 0.645 0.005 2.500 2.508 2.521 5.40 9.53 14.48 4.98 6.26 8.19 17.93 25.37 35.28 1.517 0.600 0.006 4.700 4.704 4.722 5.20 9.07 13.92 4.97 6.19 8.09 17.76 24.72 34.47 1.883 0.747 0.007 1.700 1.701 1.723 5.10 8.67 13.59 4.95 6.10 8.01 17.56 24.13 33.89 2.323 1.035 0.016 0.900 0.900 0.924 4.95 8.40 13.40 4.92 6.02 7.93 17.37 23.60 33.49 2.777 1.419 0.051 0.300 0.300 0.3

Outdoor Temperature Outdoor Humidity Ratio Outdoor Air Enthalpy Heating Load Electrical Load (no-he

Toa, °C ωoa, g/kg hoa, kJ/kg QLs, kW EL, kW 
B-1 

B. Example calculation 

B.1 General Description 

For the purpose of an example, the rating methodology was applied to a thermal load following 

fuel cell system (Type III), which was installed in the Residential Fuel Cell Test Facility at the 

National Institute of Standards and Technology (NIST) in Gaithersburg. The system was 

designed for indoor use. Table B.1 shows the measured results for the three steady state tests at 

the maximum, midpoint, and the minimum thermal energy delivery. 

Table B.1. Average measured values over duration of each steady-state test 

From these measured values, the calculations in Table B.2 were performed according to 

Section 9.2. The calculated water use rate could not be measured as indicated. NIST does not 

have the capability to fill the water reservoir as indicated in Section 9.2.6. Therefore, the water 

use was measured during each steady state test. Because the test did not necessarily start at the 

beginning of a water fill cycle and the reservoir was not filled upon completion of the test, these 

measurements are not necessarily the actual water use rates. It is assumed that manufacturers 

would provide the independent test facility performing the actual rating with the ability to fill the 

reservoir, if one exists. 

Thermal Energy Delivery 

Expanded 

Uncertainty 

(k=2) Variable Description 

Max Mid Min 

Ein

& electrical power supplied to fuel cell system, kW 0.0 0.0 0.0 0.7 % 

Eout & electrical power supplied by the fuel cell system, kW 4.61 2.75 1.06 0.7 % 

Eaux & electrical power supplied to auxiliary devices (pumps, etc.) 

not powered by the RFCS, kW 

0.6 0.6 0.6 0.7 % 

pa absolute pressure of air surrounding the RFCS, kPa 100.0 99.0 99.9 0.3 

pf absolute pressure of fuel gas at flow measurement point, 

kPa 

102.1 101.1 102.0 0.8 % 

RHa relative humidity of air surrounding the RFCS, % 49.7 49.6 49.7 1.4 

Ta temperature of air surrounding the RFCS, °C 22.2 22.1 22.1 0.3 

Tf temperature of fuel gas, °C 27.6 26.8 27.8 0.3 

The temperature of circulating fluid at the exit from the RFCS, 

°C 

54.4 52.6 51.3 0.05 

Thi temperature of circulating fluid at the inlet to the RFCS, 

°C 

50.0 50.0 50.0 0.05 

Tv temperature of the exhaust gas, °C 44.1 44.1 44.2 0.3 

Vf

& volumetric flow rate of fuel at standard conditions (T0

,P0), 

L/s 

0.467 0.307 0.181 0.2% 

Vh

& volumetric flow rate of circulating fluid through the 

RFCS, L/s 

0.448 0.448 0.448 1.2 % 

0 Hf specific energy content of fuel gas at standard conditions 

(T0

; P0), kJ/L 

38.9 38.8 38.9 0.55 % 

XCO2 volume fraction of CO2 in dry exhaust gas 0.0046 0.0036 0.0025 1 % 
B-2

Table B.2. Calculated values over each steady state test 

B.2 Nominal performance of a thermal load following RFCS (Type III) 

According to Section 9.3.3, the nominal performance of the tested system is shown below. As 

stated above, the water use rate could not be measured as prescribed, which resulted in zero 

water use measured at the maximum thermal energy delivery. For the purpose of this example, 

the nominal water use rate will be the average of the water use rate during the midpoint and 

minimum steady state tests. 

R E& = 4.61 kW 

R F& = 18.14 kW 

R Q& = 7.01 kW 

R ma & = 0.106 kg/s 

R Vw

& = 0.0 L/s 

B.3 Part-Load Functions 

Part load functions were regressed using the procedure outlined in Section 9.4.3 using the steady 

state test data presented above. As explained above, the airflow rate function could not be 

determined, but it will be added later. The following figures show the part load measured data 

and functions. Since the unit tested was specified as an indoor system, the quadratic regressions 

were performed for part load ratio, but not ambient temperature. The water use function was 

normalized to the water use at minimum thermal energy delivery instead of the maximum as 

specified. As mentioned earlier, the measured water use at maximum thermal energy delivery 

was zero. 

Thermal Energy Delivery 

Variable Description Eqn Max Mid Min 

Expanded 

Uncertaint

y (k=2) 

F& Fuel use rate, kW 9-1 18.14 12.00 7.01 0.6 % 

E fc & Net electrical power provided by the fuel 

cell system, kW 

9-2 4.01 2.15 0.46 0.7 % 

Q fc & Steady-state thermal output rate, kW 9-3 7.19 4.11 2.04 3.5 % 

on ma & Airflow rate into the RFCS while running, 

kg/s 

9-10 0.106 0.090 0.077 5.0 % 

Vw

& Water use rate L/h 9-15 0.0 0.0 2.48 1.0 % 
B-3

Fuel Use vs. Thermal Part-Load Ratio

1.00, 1.00

0.57, 0.66

0.28, 0.39

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Thermal Part-Load Ratio

Normalized Fuel Use

e1 = 0.079

e2 = 1.148

e3 = -0.227

e4 = 0.000

e5 = 0.000

e6 = 0.000

Electrical Power vs. Thermal Part-Load ratio

0.57, 0.54

1.00, 1.00

0.28, 0.11

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

Thermal Part-Load Ratio

Normalized Electrical Power

f1 = -0.389

f2 = 1.924

f3 = -0.535

f4 = 0.000

f5 = 0.000

f6 = 0.000

( ) R F& = Zf rt ,Ta × F&

( ) a 6 a

2

4 t a 5 t

2 Zf rt ,Ta = e1 + e2rt + e3rt + e r T + e r T + e T

( ) R E& fc = Ze rt ,Ta E&

( ) a 6 a

2

4 t a 5 t

2 Ze rt ,Ta = f1 + f2rt + f3rt + f r T + f r T + f T

R F& = 18.14 kW 

R E& = 4.01 kW 
B-4

Water Flow vs. Thermal Part-Load Ratio

0.57, 0.00 1.00, 0.00

0.28, 1.00

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2

Thermal Part-Load Ratio

Normalized Flow Rate

g1 = 2.769

g2 = -7.610

g3 = 4.841

g4 = 0.000

g5 = 0.000

g6 = 0.000

Ventilation Air Usage vs.Thermal Part-Load Ratio

0.28, 0.73

0.67, 0.84

1.00, 1.00

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Thermal Part-Load Ratio

Normalized Air Flow

h1 = 0.698

h2 = 0.031

h3 = 0.271

h4 = 0.000

h5 = 0.000

h6 = 0.000

B.4 Simulated Use Data 

The hourly thermal energy delivery, electrical energy production, fuel use, and water use were 

calculated (Table B.4), according to Section 9.5.3, from the simulated use data, which was 

gathered according to Section 8.1. As an indoor system, the functional relationships 9-49 

through 9-53 were not applicable, and the hourly values for each variable will not be scaled 

according to ambient temperature. 

( ) R

w w t Ta Vw V Z r , & = &

( ) a 6 a

2

4 t a 5 t

2 Zw rt ,Ta = g1 + g2rt + g3rt + g r T + g r T + g T

( ) R*

t a

*

a

* m& a = Z r m&

( ) 2

t

*

t 3

*

2

*

t 1

*

a Z r = h + h r + h r

R* ma & = 0.106 kg/s 

R Vw

& = 2.48 L/h 
B-5

Table B.4. Hourly calculated values for simulated use test 

Domestic Hot 

Water 

Withdrawn 

from Tank 

(m3) 

Thermal 

Output from 

Fuel Cell 

(kW⋅h) 

Thermal 

Energy 

Withdrawn for 

Water Heating

(kW⋅h) 

Electrical 

Energy 

Output (kW)

Fuel Use 

(kW⋅h) 

Average Air 

Flow Rate 

(kg/h) 

Fuel Cell 

System Water 

Use 

(x 10-3 m3 Hour ) 

Vwh

* Qfc* Qwh

* E wh & * Fwh

* ma,wh & * Vw

1 0.0406 3.38 1.84 1.54 11.58 319.4 0.74 

2 0.0406 2.93 1.69 1.29 9.30 304.7 0.75 

3 0.0406 2.62 1.74 1.08 8.77 298.6 0.99 

4 0.0406 2.53 1.77 1.01 8.63 299.3 1.09 

5 0.0406 2.51 1.79 0.99 8.61 297.6 1.11 

6 0.0406 2.52 1.79 0.99 8.61 294.3 1.16 

7 0 1.65 0.00 0.43 7.00 285.4 1.46 

8 0 0.11 0.00 -0.76 1.16 314.0 0.23 

9 0 0.65 0.00 -0.42 5.77 300.2 0.75 

10 0 1.70 0.00 0.51 7.00 282.6 1.46 

11 0 0.02 0.00 -0.82 1.09 310.5 0.18 

12 0 0.60 0.00 -0.48 5.43 304.3 0.62 

13 0 1.73 0.00 0.54 6.97 282.9 1.38 

14 0 0.33 0.00 -0.59 1.72 311.1 0.47 

15 0 0.04 0.00 -0.96 2.92 293.0 0.00 

16 0 1.74 0.00 0.61 8.10 291.8 1.63 

17 0 0.79 0.00 -0.25 3.51 301.2 0.90 

18 0 -0.20 0.00 -0.92 1.37 307.0 0.00 

19 0 1.51 0.00 0.21 7.86 299.5 1.21 

20 0 1.57 0.00 0.34 6.60 300.9 1.52 

21 0 -0.17 0.00 -0.92 1.35 310.7 0.05 

22 0 0.34 0.00 -0.68 4.33 315.4 0.54 

23 0 1.82 0.00 0.64 7.11 301.0 1.47 

24 0 0.12 0.00 -0.74 0.80 317.7 0.30 

Expanded 

Uncertainty 

(k=2) 

0.0001 3.5 % 5.0 % 0.7 % 0.6 % 6.0 % 1.0 % 

B.5 Energy Ratings 

The simulated use data and part load functions can now be used to calculate the predicted annual 

thermal output, fuel use, electrical output, air flow, and water use, as described in section 10.4. 

The following text was taken directly from section 10.4.5.3 and shows Eqs. 10-129 through 
B-6

10-145 with the result for the first hour of the winter representative day in Jacksonville, FL 

shown for example. The water heating requirement as well as the corresponding electrical 

energy output, fuel use, water use, and ventilation airflow quantity are determined from the hot 

water simulated use test as shown in Table B.4. 

If the unit uses indoor ventilation air, the thermal energy that is required for heating the 

ventilation air associated with water heating, * ∆Qa,wh,i , in kJ (Btu) is given by: 

[ ( ) ]

[ ] ( )

4037 kJ 1.12 kWh

max 319.4 kg / h 1 h 1.004 kJ/ kg C 21 C 9.41 C ;0

Q max m 1h c T T ;0 pa iah oa

*

a,wh,i

*

a,wh,i

= =

= × × ⋅° × ° − °

∆ = & × × −

 (10-129) 

where: 

cpa is the constant pressure specific heat for air at assumed conditions of 5 oC (40oF) 

which is 1.004 kJ/kg-oC (0.240 Btu/lbm-oF) 

Toa is the temperature of the outdoor air at hour i, oC (oF) 

Tiah is the temperature of the indoor air at heating conditions which is 21oC (70oF) 

The thermal energy used for heating water and the associated ventilation air is subtracted 

from the rated RFCS thermal output, * Qfc,i , to determine a first approximation for the 

thermal energy available for space heating from the RFCS:

7.01 kWh 1.84 kWh 1.12 kWh 0.84 kWh 3.21 kWh

Q Q Q Q Q*

loss,i

*

a,wh,i

*

wh,i

*

fc,i

*

1av,i

= − − − =

= − − ∆ − (10-130) 

where the correction, * ∆Qa,wh,i , is zero if the unit does not use indoor air for ventilation. A 

first approximation to the thermal output of an indoor Type III RFCS for space heating is 

the minimum of the available thermal energy and the space heating load: 

Q min(Q ,Q ) min(3.21 kWh, 0.939 kWh) 0.939 kWh Ls,i

*

1av,i

*

1sh,i = = = (10-131) 

This first approximation must be corrected to reflect the increase in thermal energy 

required by the residence due to heating ventilation air required by the RFCS as it supplies 

space heating. The quantity of air is determined by the first approximation to the thermal 

part-load ratio: 

0.516

7.01 kWh

0.939 kWh 1.84 kWh 0.84 kWh

Q 1hour

Q Q Q

r R*

fc

*

loss,i

*

wh,i

*

1sh,i

1t,i = + + = ×

+ + = & (10-132) 

The first approximation to the thermal part-load ratio is used in conjuction with the partload ventilation air factor to determine the first approximation to the mass of ventilation air 

associated with space heating, * ma,sh,i & in kg (lbm): 

m Z ( ) r m 0.786 0.106 kg / h 301 kg / h R*

1t,i a

*

a

* & a,sh,i = & = × = (10-133) 
B-7

The thermal energy that is required for heating the ventilation air associated with space 

heating is, ∆Qa,sh,i, in kJ (Btu): 

[ ( ) ]

[ ] ( )

3805 kJ 1.06 kWh

max 301 kg / h 1 h 1.004 kJ/ kg K 21 C 9.41 C ;0

Q max m 1h c T T ;0 pa iah oa

*

a,sh,i a,sh,i

= =

= × × ⋅ × ° − °

∆ = & × × −

 (10-134) 

The net thermal energy for space heating supplied by the RFCS is given by: 

[( ) ]

[ ] ( )

0.939 kWh

min 7.01 kWh 1.84 kWh 1.12 kWh 1.06 kWh 0.84 kWh ;0.939 kWh

Q min Q 1h Q Q Q Q ,QLs,i

*

loss,i

*

a,sh,i

*

a,wh,i

*

wh,i

* R*

sh,i

=

= − − − −

= × − − ∆ − ∆ − &

(10-135) 

There is likely some synergy associated with combining the water heating and space 

heating loads (e.g. less cycling of the system). However, the amounts of electricity 

produced, fuel used, and water used in the process of meeting the combined water heating 

and space heating requirements cannot be less than the amounts associated with water 

heating alone as reflected in the simulated use tests. Likewise, the amounts of electricity 

produced, fuel used, and water used in the process of meeting the combined water heating 

and space heating requirements cannot be less than the amounts associated with meeting 

the sum of the water heating and space heating loads while operating in steady state mode. 

Thus, the electrical output, * E , in kWh, the fuel use, i

* F in kWh (Btu), and the water use i

* Vw,i , in L (gal) are each estimated as the maximum of the respective simulated use and 

steady values: 

[ ] *

ss,i

*

wh,i

* Fi = max F ,F (10-136) 

[ ] *

ss,i

*

wh,i

* Ei = max E ,E (10-137) 

[ ] *

ss,i

*

wh,i

* Vi = max V ,V (10-138) 

where variables associated with water heating are as previously defined and: 

* Efc,ss,i is the electrical energy produced if the RFCS operated in steady state mode to 

meet the combined water heating and space heating requirement, kWh 

* Fss,i is fuel used if the RFCS operated in steady state mode to meet the combined 

water heating and space heating requirement, kWh (Btu) 

* Vw,ss,i is the water used if the RFCS operated in steady state mode to meet the 

combined water heating and space heating requirement, L (gal) 

Values for the electricity produced, fuel used, and water used in conjunction with meeting 

the water heating requirement are determined directly from the simulated use test (Eqns. 

10-108, 10-109, and 10-110). The remaining mass and energy flows for steady state 

operation to meet the combined space heating and water heating requirements ( * Efc,ss,i , * Fss,i , 

* Vw,ss,i) are based on the gross thermal energy output of the RFCS: 
B-8

0.939 kWh 1.84 kWh 1.12 kWh 1.06 kWh 0.84 kWh 5.80 kWh

Q Q Q ∆Q ∆Q Q*

loss,i

*

a,sh,i

*

a,wh,i

*

wh,i

*

sh,i

*

fc,i

= + + + + =

= + + + +

 (10-139) 

This gross thermal output is used to determine a thermal part-load ratio: 

0.827

7.01 kWh

5.80 kWh

Q 1h

Q

r R*

*

* fc,i

t,i = = × = & (10-140) 

For operation in a steady mode to meet both water heating and space heating requirements, 

the electrical energy output, fuel use, and water use are determined by this thermal part 

load ratio: 

E Z (r ) E 1h 0.836 4.01 kWh 3.36 kWh * R*

t,i

*

e

*

fc,ss,i = × & × = × = (10-141) 

F Z (r ) F 1h 0.873 18.14 kWh 15.8 kWh * R*

t,i

*

f

*

ss,i = × & × = × = (10-142) 

V Z (r ) V 1h 0.0 0.0 L /s 0.0 L /s R*

w

*

t,i

*

w

*

w,ss,i = × & × = × = (10-143) 

The electrical energy that is used by the residence is the minimum of the electrical energy 

available from the fuel cell and the electrical energy requirement of the residence, EL,i, in 

kWh: 

E min[E ,E ] min[ ] 3.36 kWh; 0.2 kWh 0.2 kWh L,i

*

fc,i

*

i = = = (10-144) 

The remaining power (if any), * G , in kWh is supplied to i the electrical grid: 

G E E 3.36 kWh 0.2 kWh 3.16 kWh *

i

*

fc,i

*

i = − = − = (10-145) 

Using a similar procedure for each hour of the three representative days in Jacksonville, FL, the 

following cumulative/annual values are calculated. 

Table B.5. Annual/Cumulative Predicted Values for Indoor Type III RFCS in Jacksonville, FL 

Description Equation Value 

Annual electrical energy 

supplied to the residence ∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

i

24

i 1

B

i

24

i 1

A AE 91 Ei 183 E 91 E 6,071 kW⋅h

Annual electrical energy 

supplied to the utility grid ∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

i

24

i 1

B

i

24

i 1

A AG 91 Gi 183 G 91 G 4,377 kW⋅h

Annual thermal energy 

delivered for water heating ∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

wh,i

24

i 1

B

wh,i

24

i 1

A AQwh 91 Qwh,i 183 Q 91 Q 3,877 kW⋅h

Annual thermal energy 

delivered for space heating ∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

sh,i

24

i 1

B

sh,i

24

i 1

A AQsh 91 Qsh,i 183 Q 91 Q 1,985 kW⋅h

Annual fuel energy consumed ∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

i

24

i 1

B

i

24

i 1

A AF 91 Fi 183 F 91 F 68,652 kW⋅h

Annual volume of domestic 

water supplied to the fuel cell 

system 

∑ ∑ ∑ = = =

= × + × + ×

24

i 1

C

w,i

24

i 1

B

w,i

24

i 1

A AW 91 Vw,i 183 V 91 V 23,813 L