Xiao-Yen J. Wang

Glenn Research Center, Cleveland, Ohio

Advanced Stirling Radioisotope Generator (ASRG) 

Thermal Power Model in MATLAB®

NASA/TM—2012-217640

August 2012
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Xiao-Yen J. Wang

Glenn Research Center, Cleveland, Ohio

Advanced Stirling Radioisotope Generator (ASRG) 

Thermal Power Model in MATLAB®

NASA/TM—2012-217640

August 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

9th International Energy Conversion Engineering Conference (IECEC)

sponsored by the American Institute of Aeronautics and Astronautics

San Diego, California, July 31–August 3, 2011
Acknowledgments

The author thanks Paul Schmitz, Edward Lewandowski, Scott Wilson, Jeffrey Schreiber, Stacey Bagg, and Robert Cataldo at 

NASA GRC for providing critical data and technical discussions for this modeling effort; Dave Meer, Robert Christie, Daniel 

Houser, Robert Cataldo for reviewing this paper; Christopher Steffen, Duane Beach, and Craig Williams at NASA Glenn Research 

Center for supporting this effort under the radioisotope power system (RPS) program.

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subject to revision as analysis proceeds.
NASA/TM—2012-217640 1

Advanced Stirling Radioisotope Generator (ASRG)

Thermal Power Model in MATLAB®

Xiao-Yen J. Wang

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

This paper presents a one-dimensional steady-state mathematical thermal power model of the ASRG. 

It aims to provide a guideline of understanding how the ASRG works and what can change its 

performance. The thermal dynamics and energy balance of the generator is explained using the thermal 

circuit of the ASRG. The Stirling convertor performance map is used to represent the convertor. How the 

convertor performance map is coupled in the thermal circuit is explained. The ASRG performance 

characteristics under i) different sink temperatures and ii) over the years of mission (YOM) are predicted 

using the one-dimensional model. Two Stirling converter control strategies, i) fixing the hot-end of 

temperature of the convertor by adjusting piston amplitude and ii) fixing the piston amplitude, were tested 

in the model. Numerical results show that the first control strategy can result in a higher system efficiency 

than the second control strategy when the ambient gets warmer or the general-purpose heat source 

(GPHS) fuel load decays over the YOM. The ASRG performance data presented in this paper doesn't 

pertain to the ASRG flight unit. Some data of the ASRG engineering unit (EU) and flight unit that are 

available in public domain are used in this paper for the purpose of numerical studies.

1.0 Introduction

The advanced Stirling radioisotope generator (ASRG) (Refs. 1 to 3) is being developed for 

multimission applications to provide a high-efficiency power source alternative to radioisotope 

thermoelectric generators (RTGs). The ASRG efficiency could reach 28 to 32 percent, which results in 

reducing the required amount of radioisotope by roughly a factor of 4 compared to RTGs. Thus, because 

of the limited supply of Pu-238, utilization of the ASRG can extend radioisotope power available for 

future space science missions, such as deep-space missions, large planetary surface rovers, and systems in 

support of human exploration activities.

An overview of the ASRG is shown in Figure 1. The ASRG consists of two advanced Stirling 

convertors (ASCs) enclosed in the housing; each has a general purpose heat source (GPHS) attached at 

the hot end to provide the heat. A gas management valve (GMV) and pressure relief device (PRD) are 

located at the top of the housing. The housing with attached fins radiates the heat to the environment. The 

GMV is used to maintain a near-atmosphere pressure of inert gas inside the housing during ground 

operations. This gas is permanently vented to vacuum by the PRD when the ambient becomes vacuum.

The ASC control unit (ACU) is separate from the ASRG housing. It converts the AC signals from both 

ASC to 28 to 34 VDC for a typical spacecraft electrical bus. The controller is used to maintain

synchronized displacer/piston movement of the two directionally opposed Stirling convertors to minimize 

induced disturbance to the spacecraft and its precision instrumentation. 

An ASC consists of a free-piston Stirling converter and an integral linear alternator that converts the 

piston reciprocating motion to electrical power output. The Stirling engine is a heat engine that operates 

by cyclic compression and expansion of a working fluid at different temperature levels such that there is 

net conversion of heat energy to mechanical work. The GPHS provides the heat to the heater head of the 

ASC. Helium is used as the working fluid that is hermetically contained within the converter pressure 

vessel and is in the various working stages of the ASC. The displacer shuttles helium between the 

expansion space, where heat is received, and the compression space, where the heat is rejected at the cold 
NASA/TM—2012-217640 2

Figure 1.—Overview of the ASRG components

side adapter flange (CSAF) attached to the housing. The changes in pressures and volumes of the 

convertor working spaces drives a power piston that reciprocates to produce AC electrical power via a 

permanent magnet linear alternator. 

The ASRG DC power output depends on the following:

1) GPHS fuel load at the year of mission (YOM)

2) Thermal insulation loss from GPHS to the environment

3) ASC efficiency (i.e., Stirling engine efficiency × alternator efficiency, the ratio of the AC power 

output and heat going into the engine)

4) Cable/connector power loss

5) ACU power loss

In this paper, a one-dimensional steady-state thermal power model is built to predict the ASRG AC 

power output. The ASC is modeled using the Sage performance map (Ref. 4) that provides the ASC 

efficiency. With given GPHS fuel load and defined ambient conditions, the AC power output is computed 

using the one-dimensional model. Assuming the cable/connector power loss is 0.8 percent of the AC 

power and the ACU efficiency is 87 percent, the ASRG DC power can be computed and will be presented 

in the paper.

In the following, the ASRG thermodynamics and energy balance will be explained first, then ASRG 

performance under varying ambient sink temperature (for different missions) and over the year of mission 

(for GPHS decay) are shown using two ASC control strategies: (i) fixing the hot-end temperature of the 

ASC and (ii) fixing the piston amplitude; which is followed by the summary and conclusions. 

2.0 Thermal Dynamics and Energy Balance of the ASRG

2.1 The Fundamentals of the Idealized Stirling Cycle

The idealized Stirling cycle is a Carnot cycle (see Figure 2) with four thermodynamic processes

acting on the working fluid:

1) Isothermal expansion at the hot-end temperature of the ASC, Th

2) Constant-volume heat removal

3) Isothermal compression at the cold-end temperature of the ASC, Tc

4) Constant-volume heat addition. 
NASA/TM—2012-217640 3

Figure 2.—P-V diagram of the Carnot cycle

The heat going into the engine, Qin, is the area of ABFE. The heat rejected from the engine, Qrej, is the 

area of CDEF. The mechanical work, W (= Qin – Qrej), is the area of ABCD. The net power output, P = W 

f with f being the frequency of the Stirling engine. It was shown in (Ref. 5) (Eq. (14)) with K = 0, Th= T3,

Qin = Q3-4, Equation (10) with K = 0, Tc = T1 ,Qrej = – Q1-2), we can have

h

D

D P

in T

V

V V Q mR + = ln (1)

c

D

D P

rej T

V

V V Q mR + = ln (2)

and

( ) ( ) h c

D

D P in rej T T

V

V V P Q Q f fmR − + = − = ln (3)

where VD is displacer swept volume, VP is the piston swept volume, m is the mass of the working fluid, 

and R is the gas constant. From Equations (1) and (2), then we can have the resistances at the hot and cold 

ends, respectively, of an idealized Stirling engine defined as

D

in D P

i h

h

V

V V fmR Q

T R

+ = =

ln

1

6 _ (4)

D

rej D P

i c c

V

V V fmR Q

T R

+ = =

ln

1 6 _

(5)

However, in practice, it is difficult to determine VD and VP. Some uncertainties of the Stirling engine 

performance such as regenerator effectiveness, dead volumes, and other losses have to be considered. The 

shaded area inside ABCD in Figure 2 represents the actual work done by the converter. West (Ref. 6) gave 

an engineering model to compute the power output of a Stirling engine as follows:

( ) h c

h c

P m n T T

T T

fV p P w − + = (6)

where pm is effective mean pressure of the Stirling engine and wn is West number for the Stirling engine, 

in the range of 0.25 to 0.35.

Following Equations (4) and (5), we can define the resistance based on the West equation as follows:
NASA/TM—2012-217640 4

h c

p m

n

w

c

w

h

T T

fV p

w

R R

+

= = 1

6 _ 6 _ (7)

It can be seen that the resistance at both the hot and cold ends are the functions of the Stirling engine

frequency, piston swept volume (piston amplitude), effective mean pressure, Th, and Tc. When the piston 

amplitude decreases, the resistance will increase. The resistances at the hot and cold ends are the same

based on equations for an idealized Stirling engine or West equation. However, in reality, the resistance at 

the hot and cold ends of the engine could be different due to the losses in the engine are counted in 

different ways at the hot and cold ends, which is shown in the Sage performance map (Ref. 4).

2.2 Thermal Circuit of the ASRG

The thermal circuit for half of the ASRG is shown in Figure 3, where Ts is the temperature of the GPHS,

Th is the temperature at the hot end of the ASC, Tc is the temperature at the cold end of the ASC, Tw is the 

housing wall temperature and Ta is the equivalent ambient sink temperature. R1 is the contact and 

conduction resistances between GPHS and housing wall; R2 is the contact and conduction resistances 

between GPHS and heat collector of the ASC; R3 is the contact and conduction resistances between the ASC 

hot end and housing wall; R4 is the conduction resistance between the cold end of the ASC and housing 

wall; R4

′ is the convection and/or radiation resistances between housing wall and the ambient environment; 

R5 is the conduction resistance between the hot and cold ends of the ASC; R6_h is the resistance of the ASC 

at the hot end. The convertor acts like a thermal resistance in the thermal perspective.

With Qtotal and Ta being defined and imposed as boundary conditions, the total heat Qtotal provided 

from the GPHS will split into Qloss_1(loss through the thermal insulation around the GPHS) and Qin to the 

ASC. The heat going into the ASC further splits into three paths: Qloss_2 (loss through conduction), Qloss_3

(loss through insulation), and Qwork ( heating up the gas to do the work). Qrej is the heat rejected from the 

ASC through the CSAF attached to the housing. All the waste heat will be rejected to the ambient through 

radiation and/or convection. How the heat splits depends on the thermal resistances in the circuit, such as 

R1, R2, R3, R4, R5, R4

′ and R6_h (see Figure 3). R6_h and Qrej are functions of Th, Tc, Qin, Qrej, and Ap (piston 

amplitude). R1, R2, R3, R4, R5, R4

′ can be defined based on the material properties and the condition of 

contact surfaces. R6_h and Qrej will be defined based on the ASC performance map (Ref. 4). Tb is a 

boundary node used to define R6_h between Th and Tb. Qrej is imposed as a heat load at a boundary node 

that has temperautre of Tc. An iterative procedure has to be used to get the correct R6_h and Qrej that satisfy 

both the energy balance of the ASRG thermal circuit and the ASC performance map. The three steps in

the iterative procedure are listed as follows; (i) give initial values of R6_h and Qrej; (ii) run the thermal 

model to compute temperatures and heat flows, such as Th , Tc , and Qin; (iii) use Th , Tc , and Qin to update 

the R6_h and Qrej based on the ASC performance map; The three steps are repeated until the differences of 

R6_h and Qrej between two iterations are small enough (10-4

) to conclude that the solution is converged. 

Note that housing temperature can be quite different on the housing and the convective heat transfer 

could vary significantly around housing when it exists. Here, the variation of the housing temperature is 

not considered. It is assumed that only radiation exists in the following ASRG performance prediction.

In the following, the relationship between Qin, Th and R6_h (i.e., Ap ), Ta at the hot end will be derived 

first, which is followed by the relationship between Qrej, Tc and R6_c (i.e., Ap ), Ta at the cold end. 

Assuming Qin = Qwork to simplify the derivation, we have

Qtotal = Qloss _1 + Qin (8)

1

_1 R

T T Q s w loss

− = (9)
NASA/TM—2012-217640 5

Figure 3.—Thermal circuit for half of the ASRG

R2

T T Q s h

in

− = (10)

h

h

in R

T Q

6 _

= (11)

Equations (10) and (11) give

Ts = QinR2 +Th (12)

Th =QinR6 _ h (13)

Substituting Equations (12) and (13) into Equation (9), we will get 

( )

1

2 6 _

_1 R

R R Q T

Q h in w

loss

+ − = (14)

By substituting Equation (14) into Equation (8), we will get

1

2 6 _

1

1

R

R R

R

T Q

Q

h

w total

in +

+

+

= (15)

Then substitute Equation (15) into Equation (13), and we get

6 _ 1

2 1

1

1 / 1

R R

R R

R

T Q

T

h

w total

h

+

+

+

= (16)

It can be seen that if Tw increases (i.e., Ta increases), Th and Qin will increase and Qloss_1 will decrease. If

R6_h is increases, Th will increase, Qin will drop, thus Qloss_1 will increase. 

At the cold end of the ASC, we have
NASA/TM—2012-217640 6

4

2

4 4

1 ' ( ) ( ) R

T T Q A T T hA T T w a rej w a w a

− = εσ − + − = (17)

R4

T T Q c w rej

− = (18)

c

c rej R

T Q

6 _

= (19)

where A1 is the housing area, A2 is the convective heat transfer area, ε is the surface emissivity, σ is the 

Stefan-Boltzmann constant, and h is the convection heat transfer coefficient. Equations (17) and (18) give 

Tw Ta R Qrej

' = + 4 (20)

Tw = Tc − R4Qrej (21)

By combining Equations (19), (20), and (21), we get

c

a c

R

R R

T T

6 _

4 4

' 1 + −

= (22)

6 _ 4

4

' 1

R R

R

T T

c

a w

− −

= (23)

It can been seen that when Ta increases, Tc and Qrej will increase. When R6_c increases, Tc and Qrej will 

decrease. 

In summary,

1) If the piston amplitude decreases, 

o R6_h and R6_c increase, then 

o Th increases; Equation (16)

o Qin drops; Equation (15)

o Tc drops; Equation (22)

o Qrej drops; Equation (19)

o the net power output (P) drops.

2) If the sink temperature increases, 

o Th increases; Equation (16)

o Qin increases; Equation (15)

o Tc increases; Equation (22)

o Qrej increases; Equation (19)

o the net power output (P) drops.

3.0 One-Dimensional Thermal Power Model Results

The one-dimensional thermal model described above is built in Matlab and used to predict the ASRG 

performance using two ASC control strategies. In the following, the ASRG performance characteristics 

under YOM is described first, which is followed by the corresponding performance under different 

ambient sink temperature. 
NASA/TM—2012-217640 7

3.1 ASRG Performance Characteristic Under YOM

The ASRG power output is studied for the YOM since the GPHS fuel decays with time. A 17-year 

time frame that includes up to 3-year storage and 14-year on mission is considered, and the fuel decay 

rate is computed as

87.74 YOM

YOM GPHSfuel load = BOM fuel load×0.5

The begining of mission (BOM) fuel load is 244, 250, and 256 W for minimum, nominal, and maximum 

cases, respectively. As shown in Figure 4, for BOM fuel load = 244 W, each GPHS fuel load drops 1.8 W 

per year and has a 30.6-W total drop at the end of mission (EOM). For BOM fuel load = 256 W, each 

GPHS has a 32.0-W total drop at the EOM. The fuel almost drops linearly with time. 

With the fuel decaying over the YOM, the ASRG power output in deep space (no sun, Ta = 4 K) is 

shown in Figure 5(a) using the first ASC control strategy(fixed Th). It can be seen that ASRG DC power 

output drops 19 W within the 17-year period; that is, a 1.12-We drop per year. Qin and Qrej are decreasing 

since the total heat is decreasing as shown in Figure 5(b) for one ASC; while Th is maintained as constant

as that at the BOM, Tc decreases a little as shown in Figure 5(c), which results in a higher ASC efficiency. 

The piston amplitude at EOM is approximately 91.1 percent of that at the BOM. 

For the case of starting from the minimum fuel load at BOM, the ASRG DC power output is 121-We 

with the ASRG efficiency of 28.4 percent at EOM, compared with a 140-We DC power output with 

28.6 percent ASRG efficiency at BOM in deep space, which is referred as the current best estimation 

(CBE) ASRG power output in (Ref. 7). 

The corresponding ASRG power output with the second ASC control strategy(fixed piston amplitude) 

is presented in Figure 6, showing ASRG DC power output drops 30.5 We within the 17-year period 

starting from the minimum fuel load at BOM. Qin and Th are decreasing with YOM, and Qrej and Tc

remain almost constant. The ASC efficiency drop is more significant; thus power output drops more. At 

EOM, the ASRG DC power output is 109.5-We with the efficiency of 25.7 percent.

For the case of starting from the maximum fuel load at BOM, the ASRG output power drop is very 

similar to that starting at the minimum fuel load. Also we can see that the DC power output is decreasing 

linearly with YOM since the fuel decay is almost linear. 

Figure 4.—The GPHS fuel decay under YOM.
NASA/TM—2012-217640 8

(a)

(b) (c)

Figure 5.—ASRG performance under YOM with fixing Th.
NASA/TM—2012-217640 9

(a)

(b) (c)

Figure 6.—ASRG performance under YOM with a fixed piston amplitude.

3.2 ASRG Performance Characteristic Under Different Ambient Sink Temperature

For different missions, the environment could be very different. Here, the ambient sink temperature

Ta varying from 4 to 300 K as shown in (Ref. 7) is investigated. The equivalent sink temperature of 200 K 

is used for the environment on Mars. For a Venus flyby, the sink temperature can be 300 K. It is assumed 

here that heat only radiates between the ASRG housing and the ambient. The GPHS fuel load is assumed 

to be 244, 250, and 256 W for minimum, nominal, and maximam cases at BOM, respectively. The two 

ASC control strategies are also investigated here.

Figure 7 shows how the ASRG power output varies with the sink temperature for the first ASC 

control strategy. When the sink temperature gets higher—that is, a warmer environment—Qin increases a 

little because insulation loss decreases shown in Figure 7(b) for one ASC. If the Th is fixed, Tc increases
NASA/TM—2012-217640 10

while Ta gets higher as shown in Figure 7(c), which results in a lower ASC efficiency. Thus the power 

output will drop. For the minimum fuel load case, at Ta = 200 K, we can see a 5.2 We power drop from Ta 

= 4 K, while at Ta = 300 K, we can see that 21-We drop and the ASRG efficiency becomes 24.4 percent. 

In order to maintain Th constant, the piston amplitude at Ta = 300 K becomes 97.8 percent of that at 

Ta = 4 K.

Figure 8 shows the corresponding result using the second ASC control strategy. For the minimum 

fuel load case, the ASRG DC power output drops 7 We when Ta = 200K, and 27 We drop when 

Ta = 300 K. If the piston amplitude is fixed as a constant, Th drops and Tc increases while Ta gets higher as 

shown in Figure 8(c), which results in a larger ASC efficiency drop. At Ta = 300 K, the ASRG efficiency 

becomes 23.2 percent.

It shows that the first ASC control strategy can achieve higher ASC efficiency when the ambient gets 

warmer, thus the power drop becomes less than that when using the second ASC control strategy. 

(a)

(b) (c)

Figure 7.—ASRG performance under varying sink temperature Ta with fixing Th.
NASA/TM—2012-217640 11

(a)

(b) (c)

Figure 8.—ASRG performance under varying sink temperature Ta with a fixed piston amplitude.

4.0 Conclusions

A one-dimensional steady-state mathematical thermal power model of the ASRG has been presented. 

How the ASC represented by Sage performance map is coupled in the ASRG thermal circuit has been 

explained. The ASRG performance characteristics under different environments and GPHS fuel load 

decay with YOM have been shown. Further validation of the one-dimensional model with a threedimensional finite element ASRG thermal-power model for the flight unit or EU will be performed in

future work.
NASA/TM—2012-217640 12
NASA/TM—2012-217640 13

Appendix—Symbols List

A1 housing surface area

A2 heat transfer area

Ap piston amplitude

f frequency of the Stirling engine

h convection heat transfer coefficient 

m mass of the working fluid 

P the power output of Stirling engine

pm mean pressure of the ASC

Qin heat going to the ASC

Qloss_1 heat loss from the GPHS through insulation material to the ambient

Qloss_2 heat loss from the ASC hot end through insulation material to the ambient

Qloss_3 heat loss through conduction inside the ASC

Qrej heat rejected from the cold end of the ASC

Qtotal total heat generated from the GPHS

Qwork heat going to do the work

R gas constant 

R1 contact and conduction resistances between GPHS and housing wall

R2 contact and conduction resistances between GPHS and heat collector of the ASC

R3 contact and conduction resistances between the ASC hot end and housing wall

R4 conduction resistance between the cold end of the ASC and housing wall

R4

′ convection and/or radiation resistances between housing wall and the ambient environment

R5 conduction resistance between the hot and cold ends of the ASC 

R6_h resistance at the hot end of the ASC

R6_c resistance at the cold end of the ASC 

i R6 _ h resistance at the hot end of an idealized Stirling engine

i R6_ c resistance at the cold end of an idealized Stirling engine

w R6 _ h resistance at the hot end based on the West equation

w R6 _ c resistance at the cold end based on the West equation

Ta equivalent ambient sink temperature

Th temperature at the hot end of the ASC

Tc temperature at the cold end of the ASC

Ts temperature of the GPHS 

Tw housing wall temperature 

VD displacer swept volume 

Vp piston swept volume

W the mechanical work from Stirling engine

wn West number

ε surface emissivity

σ Stefan-Boltzmann constant
NASA/TM—2012-217640 14

References

1. J. Chan, J.G. Wood, J.G. Schreiber, “Development of Advanced Stirling Radioisotope Generator for 

Space Exploration,” presented at the Space Technology and Applications International Forum 

(STAIF), February 11-15, 2007.

2. E.J. Lewandowski, J.G. Schreiber, “Testing to characterize the Advanced Stirling Radioisotope

Generator engineering unit,” presented at the 8th International Energy Conversion Engineering 

Conference (IECEC), July 25-28, 2010.

3. T.F. Regan, E.J. Lewandowski, “Development of a Stirling system dynamic model with enhanced 

thermodynamics,” NASA/CR—2005-214018.

4. Sage data of ASC E2, private communication with G. Wood at Sunpower, May 13, 2009.

5. B. Kongtragool, S. Wongwises, “Thermodynamic analysis of a Stirling engine including dead 

volumes of hot space, cold space and regenerator,” Renewable energy, Volume 31, Issue 3, March 

2006, pp. 345–359, Pascal Puech, Victoria Tishkova.

6. C.D. West, “Stirling Engines and Irrigation pumping,” ORNL/TM-10475, Sep. 8, 1987.

7. ASRG User ICD, 912IC002085, Lockheed Martin, May 2010.

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Advanced Stirling Radioisotope Generator (ASRG) Thermal Power Model in MATLAB® 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

This paper presents a one-dimensional steady-state mathematical thermal power model of the ASRG. It aims to provide a guideline of 

understanding how the ASRG works and what can change its performance. The thermal dynamics and energy balance of the generator is 

explained using the thermal circuit of the ASRG. The Stirling convertor performance map is used to represent the convertor. How the 

convertor performance map is coupled in the thermal circuit is explained. The ASRG performance characteristics under i) different sink 

temperatures and ii) over the years of mission (YOM) are predicted using the one-dimensional model. Two Stirling converter control 

strategies, i) fixing the hot-end of temperature of the convertor by adjusting piston amplitude and ii) fixing the piston amplitude, were tested 

in the model. Numerical results show that the first control strategy can result in a higher system efficiency than the second control strategy 

when the ambient gets warmer or the general-purpose heat source (GPHS) fuel load decays over the YOM. The ASRG performance data

presented in this paper doesn't pertain to the ASRG flight unit. Some data of the ASRG engineering unit (EU) and flight unit that are 

available in public domain are used in this paper for the purpose of numerical studies.

15. SUBJECT TERMS

Thermal modeling; Stirling radioisotope generator 

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