Roy C. Tew, Jr.

Glenn Research Center, Cleveland, Ohio

Mounir B. Ibrahim

Cleveland State University, Cleveland, Ohio

Study of Two-Dimensional Compressible

Non-Acoustic Modeling of Stirling

Machine Type Components

NASA/TM—2001-211066

July 2001

IECEC2001–CT–27
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Glenn Research Center

Prepared for the

36th Intersociety Energy Conversion Engineering Conference

cosponsored by the ASME, IEEE, AIChE, ANS, SAE, and AIAA

Savannah, Georgia, July 29–August 2, 2001

Roy C. Tew, Jr.

Glenn Research Center, Cleveland, Ohio

Mounir B. Ibrahim

Cleveland State University, Cleveland, Ohio

Study of Two-Dimensional Compressible

Non-Acoustic Modeling of Stirling

Machine Type Components

NASA/TM—2001-211066 IECEC2001–CT–27
Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

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endorsement, either expressed or implied, by the National

Aeronautics and Space Administration.

Available electronically at http://gltrs.grc.nasa.gov/GLTRS
NASA/TM—2001-211066 1

Proceedings of IECEC’01

36th Intersociety Energy Conversion Engineering Conference

July 29–August 2, 2001, Savannah, Georgia

IECEC2001–CT–27

STUDY OF TWO-DIMENSIONAL COMPRESSIBLE NON-ACOUSTIC MODELING OF

STIRLING MACHINE TYPE COMPONENTS

Roy C. Tew, Jr.

Thermo-Mechanical Systems Branch

Power and On-Board Propulsion Division

National Aeronautics and Space Administration

Glenn Research Center, Mail Stop 301–2

Cleveland, Ohio 44135–3191

Phone: 216–433–8471, Fax: 216–433–8311

E-mail: Roy.Tew@grc.nasa.gov

Mounir B. Ibrahim, Chairman and Professor

Mechanical Engineering Department

Fenn College of Engineering

Cleveland State University

1960 East 24th Street

Cleveland, Ohio 44115–2425

Phone: 216–687–2580, Fax: 216–687–9280

E-mail: m.ibrahim@csuohio.edu

ABSTRACT

A two-dimensional (2-D) computer code was developed for

modeling enclosed volumes of gas with oscillating boundaries,

such as Stirling machine components. An existing 2-D

incompressible flow computer code, CAST, was used as the starting

point for the project. CAST was modified to use the compressible

non-acoustic Navier-Stokes equations to model an enclosed

volume including an oscillating piston. The devices modeled have

low Mach numbers and are sufficiently small that the time required

for acoustics to propagate across them is negligible. Therefore,

acoustics were excluded to enable more time efficient computation.

Background information about the project is presented. The

compressible non-acoustic flow assumptions are discussed. The

governing equations used in the model are presented in transport

equation format. A brief description is given of the numerical

methods used. Comparisons of code predictions with experimental

data are then discussed.

INTRODUCTION

Background

NASA Glenn Research Center (GRC) has been involved in

development of Stirling engines for ~25 years. GRC began

managing the Stirling Automotive Development Program for the

Department of Energy (DOE) in ~1977. The DOE/NASA

contractors were first Ford Motor Co. (for one year) and then

Mechanical Technology, Inc. (MTI) of Albany, NY. These Stirling

automotive engines used hydrogen as the working fluid and were

required to operate over demanding automotive driving cycles.

This work continued into the early ‘90’s (Ernst and Shaltens [1]).

Engines were demonstrated in automobiles and trucks and future

development looked promising. However, due to decreases in oil

and gasoline prices and improvements in spark ignition engines,

the automotive manufacturers did not choose to develop the

Stirling technology for the automotive market.

Development of Stirling engines for generation of space

auxiliary power began in ~1983 with MTI as the primary

contractor. These engines used helium as working fluid, were

25 kWe (12.5 kWe/cylinder) designs with electrical power

produced by linear alternators and used gas bearings. Back-to-back

cylinders with synchronized pistons damped engine vibrations.

One of the problems encountered in Stirling engine

development by GRC and MTI was the limited accuracy of the 1-D

engine design computer codes. Performance projections for new

designs were usually optimistic. The last major design of a 25 kWe

(12.5 kWe/cylinder) Stirling engine for space power was the

Component Test Power Converter or CTPC (Dahr [2]). Because of

previous power shortfalls with other new engine designs, MTI and

NASA chose to design the CTPC with a 20% margin on power.

When tested the engine produced slightly in excess of the 25 kWe

power goal, meaning that it produced almost 20% less power than

predicted by the computer code that was used to design it.

During this period of space power Stirling engine

development at MTI (~1983 to 1993), concerns about 1-D design

code accuracy led to fundamental research. A “loss understanding”

program was started consisting of grants and small contracts for

investigation of thermodynamic losses (Tew and Geng [3]). The

participants were hosted by GRC at yearly Stirling Loss

Understanding Workshops for several years. The long range goal

was to improve the accuracy of 1-D Stirling engine design codes.

The SP-100 (or Space Power-100 kWe) reactor space power

system program, for which these 25 kWe space power Stirling

designs were intended, ended in 1993. Along with it, funding for

the Stirling loss understanding program ended also.

From ~1993 to 1999, NASA support of Stirling engine

development continued at a low level via NASA SBIR funding.

Most of this funding was for continued development of free-piston

engines at the Stirling Technology Company (STC) of Kennewick,

WA based on flexural bearing technology. GRC and Cleveland

State University (CSU) also continued low-level development of a

multi-D Stirling code via the thesis project reported here.

Now Stirling power convertors are being developed by DOE,

NASA and STC as a substitute for the less efficient radioisotope

thermoelectric generators for deep-space missions (Thieme, Qui, and

White [4]). The success of STC in accumulating years of reliable

operation on its free-piston flexural bearing machines (over 6 years

on a 10 kWe convertor with no maintenance and no degradation in

performance) has been a key factor. In late 1999, a 55 We STC freepiston Stirling was subjected to a number of tests and evaluations

to determine its suitability for a deep-space mission (For example, it

passed a simulated launch vibration test, while operating). As a

result, in 2000, the Stirling radioisotope power system, or RPS

(Furlong and Shaltens [5]), was baselined as the advanced power
NASA/TM—2001-211066 2

system that may be used on NASA missions such as Europa and

Solar Probe missions. It now appears that long-duration Martian

rover missions will be the first application of Stirling RPS.

Stirling One-Dimensional Flow Design Codes

Two one-dimensional-flow Stirling machine codes have been

in primary use by NASA’s Stirling engine contractors in recent

years. These are the HFAST code developed by Mechanical

Technology, Inc. (Huang [6]) and the GLIMPS code (now Sage,

(Gedeon [7])) developed by Gedeon Associates. HFAST was used

by MTI in design of the large (25 kWe) Stirling convertors

discussed earlier. The GLIMPS code and now its successor, Sage,

have been the primary design codes used by the Stirling

Technology Company. Sage is also used by Sunpower, Inc. While

these codes have been used to develop excellent engine designs, 1-

D codes assume uniform axial flow and are thus believed to be

deficient in modeling those interfaces in Stirling engines where

significant changes in area take place. Past comparisons of code

predictions, such as the GLIMPS/HFAST study done by Geng and

Tew [8], showed rough overall performance agreement but

differences in predictions of individual losses. It would be of

significant value to be able to accurately characterize all major

losses so that the design process could do an adequate job of

trading off the various losses against each other.

Objective of the Work

The immediate purpose of the 2-D code development reported

here was to develop a time-efficient compressible code for study and

definition of Stirling machine type cylinder heat transfer/power (or

hysteresis) losses; see Tew [9] for more complete documentation. A

long range purpose was to provide the basis for development of a 2-

D model of a complete Stirling engine that could be used to study

interaction of various thermodynamic losses. Even longer range, if

computer software/hardware time efficiencies are increased

sufficiently, it may be possible to use a 2-D code (perhaps even a 3-D

code) for engine design. The previous IECEC paper by Makhkamov

and Ingham [10] suggests such design use of a 2-D code is already

underway at the Laboratory for Stirling Engines at the PhysicalTechnical Institute in Tashkent.

NOMENCLATURE

Latin letters

cp specific heat at constant pressure

 d denotes differential of variable

Dh hydraulic diameter (=4 x wetted area/wetted perimeter)

D viscous force vector

f mass force vector (due to gravity, for example)

p static pressure

P mean spatial pressure

Pa amplitude of mean spatial pressure

Po arithmetic mean of max. and min. mean spatial pressures

qm entropy source due to non-zero mass sources

Q density of continuously distributed heat sources

R gas constant

s entropy per unit mass

t time

T temperature

T

ST stress tensor

u velocity vector (= iU jV + )

U axial, x-direction, velocity

V radial, r-direction, velocity or volume

Vo arithmetic mean of maximum and minimum volumes

w u = ξρ rate of momentum change because of mass sources

Wˆ

loss non-dimensional work or hysteresis loss

Greek letters

γ ratio of specific heats of fluid

φ transport quantity per unit mass

λ molecular thermal conductivity

μ absolute viscosity

ρ density, fluid mass per unit volume

ω angular velocity (rad/sec)

ξ mass source strength per unit mass

COMPRESSIBLE NON-ACOUSTIC FLOW

Assumptions

Fedorchenko [11] discusses a number of subsonic initialboundary-value problems that cannot be solved using the classical

theory of incompressible fluid motion, which involves the equation

∇⋅ = u 0 (where u iU jV = + and the unit vector i is the xdirection for both cartesian and axisymmetric coordinates; the unit

vector j is the y-direction for cartesian coordinates and in the rdirection for axisymmetric coordinates.) Among these problems are:

(1) flows in a closed volume initiated by blowing or suction

through permeable walls, (2) flows in a closed volume initiated by

a moving boundary such as in a piston-cylinder problem, (3) flows

with continuously distributed mass sources, and (4) viscous flows

with substantial heat fluxes. Fedorchenko notes that application of

the most general theory of compressible fluid flow may not be best

in such cases because of the difficulties in accurately resolving

complex acoustic phenomena and in assigning proper boundary

conditions.

Fedorchenko proposes a non-local mathematical model where

∇⋅ ≠ u 0 in general, for simulation of unsteady subsonic flows in a

bounded domain with continuously distributed sources of mass,

momentum and entropy, also taking into account the effects of

viscosity and conductivity when necessary. The exclusion of

sound waves is one of the most important features of the model.

The most general form of Fedorchenko’s compressible nonacoustic system of equations for simulation of unsteady subsonic,

heat conducting viscous flows are the following forms of the

momentum, continuity, energy and state equations:

∂ ρ

∂

ρ ρ

( ) u

t

+∇⋅ +∇ = + + uu p f w D (1)

∂ρ

∂

ρ ξρ

t

+∇⋅ = u (2)

∂

∂

λ s

t

u s R

P

+ ⋅∇ = ∇⋅ ∇ + + [ ] TQqm (3)

FsP ( ) , , ρ = 0 (4)

When ∇ = μ 0 (as it is for constant viscosity problems) then

Du u = + ∇ ∇⋅ μ μ ∆

1

3

( ) (5)

Note that the energy equation is written in terms of entropy, s,

rather than in terms of internal energy, enthalpy, or temperature.
NASA/TM—2001-211066 3

The simplifications in the Navier-Stokes equations used to

eliminate acoustics and arrive at the above system of equations

were: (1) Pressure at any time, t, and position, r, is split into a

mean spatial pressure level that varies only with time and a

∆( ) pressure relative to a reference position that varies with

position and time:

prt Pt prt ( ,) () ( ,) = +∆ (6)

(2) The pressure appearing in the equation of state is the mean

spatial pressure that varies only with time. Therefore, from the ideal

gas equation of state

ρ = P t

RT r t

( )

(,)

(7)

So density is a function of mean spatial pressure level and the

temperature field. (and is independent of the spatial pressure drop).

Simplification of Equations for the Stirling Problem

For the Stirling piston-cylinder problem of interest here, there

are no mass or heat sources distributed within the cylinder volume

and the gravity force on the gas is not of interest. Therefore the

variables defined above: f w, , ξ , Q and qm are all zero. Also the

ideal gas equation of state is sufficiently accurate for the helium gas

that is used in most current Stirling engines of interest to NASA.

Therefore the equations (1)-(4) reduce to the following set:

∂ ρ

∂

ρ

( ) u

t

+∇⋅ +∇ = uu p D (8)

∂ρ

∂

ρ

t

+∇⋅ = u 0 (9)

∂

∂

λ s

t

u s R

P

+ ⋅∇ = ∇⋅ ∇ [ ] T (10)

ρ = P

RT

(11)

CAST AND MODIFIED CAST CODE

The incompressible flow CAST ( C omputer A ided S imulation of

T urbulent Flow) code, as originally received by Cleveland State

University and NASA is documented in Peric and Schuerer [12].

Changes made to CAST in developing the modified CAST

compressible non-acoustic code are documented in Tew [9]. 

Transport Equation Format

The solution technique used in the CAST code is based on a

transport equation formulation of the governing equations. A

coordinate-free form of the general transport equation is:

∂ ρφ

∂

ρ φ φ φ φ

( ) +∇⋅ − ∇ = ( ) t

u S Γ (12)

The text under equation (12) categorizes the terms in the equation.

Transport quantities, exchange coefficients and source terms for

the continuity, momentum and energy equations used in Modified

CAST are shown in Table I. Note that the energy equation is in

enthalpy format, rather than the entropy format used by

Fedorchenko. The turbulent kinetic energy and dissipation rate

equations are not shown here due to space limitations. They are

defined in Tew [9].

Table I: General Transport Equation—Transported Quantities, Exchange

Coefficients, and Source Terms for Continuity,

Energy and Momentum Equations

Equation

Name

Transport

Quantity/vol.

Exchange

Coefficient

Source

Term

Continuity ρ (mass/vol.) 0 0

Momentum ρu (momentum/vol) μ ∇ ⋅ T

ST S,

Energy ρh (enthalpy/vol.) λ μ

cp

=

Pr

dP

dt

Numerical (Finite-Volume) Methods

The numerical methods in modified CAST are almost the same

as those used in CAST (Peric and Scheuerer [12]). An exception is

the use of Leibniz’s rule (Ferziger and Peric [13]) to account for the

effect of the moving boundary (i.e., piston motion). Modifications to

the governing equations themselves, to account for the change from

incompressible to compressible non-acoustic flow, are discussed in

Tew [9]. These included accounting for the non-zero ∇ ⋅ u terms,

adding the time derivative of the mean spatial pressure to the source

terms of the energy equation, and using mean spatial pressure in the

ideal gas equation so that the density field is a function of the

temperature field and the mean spatial pressure. Spatial pressure

drop still appears in the momentum equations to help determine the

velocity field. Since the density is not affected by spatial pressure,

the incompressible SIMPLE algorithm still applies.

The governing equations are solved with a conservative finitevolume method (see Patankar [14]). The basic principle is:

(1) Discretize the solution domain by subdividing it into small

axisymmetric (or rectangular) control volumes. Locate the numerical

grid points in the center of the control volumes.

(2) Discretize the transport equations.

Discretization is done by formally integrating the single terms

in the equations over a control volume. Application of Leibniz’s

rule and Gauss’s theorem yields an integro-differential equation

relating the net increase in the transported quantity per unit time to

the convective and diffusive fluxes across the control volume

boundaries and to the source (or sink) terms within the control

volume. This practice leads to a conservative method because

boundary-fluxes leaving one control volume through its right

boundary enter the neighboring control volume through its left

boundary. Since this principle applies to all control volume faces,

the scheme becomes overall conservative. The approach is

described in Peric and Schuerer [12] and Tew [9] in more detail.

AVAILABLE DATA FOR CAST COMPARISONS

Rectenwald Computations and Kornhauser

Experimental Data

Recktenwald [15] computed heat transfer between the walls

and gas inside the cylinder of a reciprocating compressor. A

compressor cylinder contains intake and discharge valves, unlike

the cylinders of a Stirling engine. Recktenwald used 2-D, unsteady,

compressible equations (acoustics included) to simulate the

compressor. In order to validate his computer code, he simulated a

gas spring for comparison with data generated by Kornhauser and

Smith [16, 17]. The comparison between data and experiment was

based solely on experimental and simulated values of nondimensional hysteresis loss for the gas spring over an operating

range. It should be noted that a gas spring is essentially a pistoncylinder which has no flow to or from the enclosed volume of the

cylinder. Kornhauser [18] reported further details of these gas

spring experiments. He also reported on tests made with a

Time rate

of storage convection diffusion source
NASA/TM—2001-211066 4

modification of the gas spring test rig, to include a heat exchanger

mounted on top of the cylinder, such that flow could continuously

pass between the cylinder and the heat exchanger as the piston

expanded and compressed the gas. This “two-space test rig”

operated more like a Stirling machine cylinder than either a gas

spring or a compressor.

Kornhauser’s gas spring test data was also used as a basis for

validation of the 2-D Modified CAST compressible non-acoustic

code. Hysteresis losses computed by CAST were compared with the

experimental values of Kornhauser and the calculated values of

Recktenwald, over a range of gas spring operation. Also, since

Recktenwald published plots of his calculated velocity vectors and

temperature contours within the gas spring for two operating

points, these were compared with similar plots of CAST calculated

values for one of the operating points. Thus it was possible to

compare the calculations of a compressible model, which did

account for acoustics in the computations against the compressible

non-acoustic calculations of the Modified CAST code. Comparison

of CAST and Reckenwald’s temperature contours over the cycle for

a 10 RPM gas spring showed excellent agreement. Agreement

between velocity vector plots also appeared to be very good,

although due to some difference in the way the plots were made

only a qualitative comparison could be made. These 2-D

comparisons are shown in Tew [9].

Gas Spring and Two-Space Test Rig Dimensions

The dimensions of Kornhauser’s test rigs are shown in Tables

I and II. Due to a limitation of the CAST code in simulating complex

geometries, it was necessary to make a slight change in the annular

heat exchanger geometry simulated for the two-space test rig. The

annular heat exchanger, physically mounted on top of the cylinder,

was moved slightly outward so that the outer wall coincided with

the cylinder wall, and the heat exchanger volume was maintained

the same in order to maintain the same volume ratio. Physical and

simulated dimensions are shown in Table III (This small change in

geometry likely tended to reduce the difference between test and

data results, to be discussed below).

Table II: Gas Spring Dimensions

Physical Quantity Symbol Value

Cylinder Bore (Diam.) D 50.80 mm (2 in.)

Piston Stroke S 76.2 mm (3 in.)

Volume Ratio r

ν

2.0

Table III: Two-Space Test Rig Dim. (Physical and Simulated)

Physical Quantity Physical Value Simulation Value

Cylinder Bore 50.80 mm (2 in) 50.80 mm (2 in)

Piston Stroke 76.20 mm (3 in) 76.20 mm (3 in)

Volume Ratio 2.0 2.0

Annulus O.D. 44.5 mm (1.75 in) 50.80 mm (2 in)

Annulus I.D. 39.4 mm (1.55 in) 46.4 mm (1.83 in)

Annulus Gap 2.5 mm (0.10 in) 2.2 mm (0.09 in)

Annulus Length 445 mm (17.5 in) 445 mm (17.5 in)

Min Pist/Head Clr. 2.9 mm (0.11 in) 2.9 mm (0.11 in)

Both test rigs used helium gas and the walls of the cylinder

and heat exchanger were at a constant temperature of approximately

294 K.

RESULTS: COMPUTATIONS VS. DATA

Gas Spring Hysteresis Losses

For a gas spring, the hysteresis loss is the work that is

dissipated by the spring per cycle at steady operating conditions;

it’s also equal to the heat generated in and transferred out of the

spring. A good way to compare computational and measured

hysteresis losses is via plots of dimensionless work as a function of

oscillating flow Peclet number. Dimensionless work and

oscillating flow Peclet numbers are defined, respectively, as

follows:

Wˆ

loss

P dV

PoVo

Pa

Po

= ∫

 −



 



 



 





2

γ 1

γ

(13)

Pe ocp Dh

ω

ρ ω

λ

=

2

4

(14)

Recktenwald had previously plotted his calculated

dimensionless losses on a plot of Kornhauser’s data. The Modified

CAST dimensionless losses were superimposed on this plot and

the result is shown in Figure 1. This figure shows dimensionless

loss as a function of oscillating flow Peclet number. The CAST loss

values were plotted at “uncorrected” Peclet numbers as the

obviously handwritten “X” symbols. If plotted at the corrected

values of the Peclet numbers they would be shifted slightly to the

right and would fall on Recktenwald’s values (the solid

diamonds).

Five of the six pairs (of CAST and Recktenwald values) of

calculated dimensionless losses agree well with Kornhauser’s data.

The one pair of calculated points that did not agree well with the data

is the pair shown at the highest Peclet number and corresponds to a

1000 RPM, 1465 kPa gas spring operating condition. Recktenwald

[15] discusses several plausible explanations for disagreement at

high Peω.

Figure 1: Modified CAST Dimensionless Losses Superimposed on

Recktenwald [15] Plot of Recktenwald's Calculated Losses and

Kornhauser's [18] Experimentally Derived Losses
NASA/TM—2001-211066 5

Also in the mid-range of the data in the vicinity of Peω=10,

where there appears to be a higher and a lower data curve, the data

agrees with the lower data curve. Kornhauser [18] found that the

higher and lower data “curves” were related to differences between

data taken at high-pressure/low-speed (higher curve) and that

taken at low-pressure/high speed (lower curve) which had

oscillating flow Peclet numbers in the same range. Thus he

concluded there was some other dimensionless parameter needed to

resolve the data in this range. His experiment also showed that

adding fins within the clearance volume of the gas spring

suppressed the difference in losses of the two types of data (highpressure/low-speed and low-pressure/high-speed). See Kornhauser

[18] for more discussion.

Comparison of modified CAST and experimental P-V diagrams

are shown in Figures 2 and 3 for ~49 and 496 RPM respectively.

Agreement is very good. When plotted in Microsoft Excel the

curves were smooth. The ‘jagged” nature of the curves appeared as

an artifact of “pasting” the Excel plots into Microsoft Word.

300.0E+3

400.0E+3

500.0E+3

600.0E+3

700.0E+3

800.0E+3

900.0E+3

1.0E+6

100 150 200 250 300 350

Volume, cm^3

PressExp.(N/m^2)

Press-CAST(N/m^2)

Figure 2: Comparison of Experimental and CAST P-V Diagrams.

CAST used 18 x 12 grids, 120 time steps/cycle

(Operating Conditions: 48.6 RPM, Mean Pressure = 555.7 kPa

(80.6 psia), Wall Temp. = 294 K)

000.0E+0

50.0E+3

100.0E+3

150.0E+3

200.0E+3

250.0E+3

300.0E+3

350.0E+3

400.0E+3

100 150 200 250 300 350

Volume, cm^3

Press-Exp(N/m^2)

Press-CAST(N/m^2)

Figure 3: Comparison of Experimental and CAST P-V Diagrams.

CAST used 18 x 12 grids, 120 time steps/cycle

(Operating Conditions: 495.8 RPM, Mean Pressure=223.3 kPa

(32.4 psia), Wall Temp. = 294 K)

Two-Space Test Rig Data and Calculation Comparisons

Figures 4 and 5 show experimental heat exchanger heat transfer

per unit area and annulus center-to-wall temperature difference for

Kornhauser’s [18] two-space rig. Figures 6 and 7 show the

corresponding Modified CAST calculated results for comparison

with Figs. 4 and 5. The experimental and calculated heat transfer

have different signs due to differences in definition of the positive

heat transfer direction.

Figure 4: Heat Transfer per Unit Area at Various Positions in Heat

Exchanger Relative to Entrance to Cylinder. Kornhauser [18]

Exp. Data: Run #12071539, 201.7 RPM,

1.008 MPa Mean Pressure

Figures 4 and 6 show that peak experimental heat transfer/unit area

near the entrance is 4000 to 5000 W/m2

 less than the

corresponding calculated value; near the end of the heat exchanger

the peak experimental value is about 9000 W/m2

 less than the

corresponding calculated value. Thus even though the qualitative

variations in heat transfer look very similar in the experimental and

calculated plots, the quantitative agreement is not very good.

Comparison of the annulus center to wall temperature

differences in Figs. 5 and 7 show that the calculated temperature

differences are smaller that the experimental values. This is

consistent with the calculated heat transfers/area being larger than

the experimental values.

Two-dimensional plots of the calculated temperatures,

velocities, pressures, etc. are given in Tew [9]. However, there are

no experimental values available for comparison.

Figure 5: Temperature Difference from Heat Exchanger Center

to Wall. Kornhauser [18]

Exp. Data: Run #12071539, 201.7 RPM,

1.008 MPa Mean Pressure
NASA/TM—2001-211066 6

Figure 6: Modified CAST Calculation of Heat Transfer per Unit Area

at Various Positions in Heat Exchanger Relative to Entrance to

Cylinder (34x20 grids, 120 time steps/cycle). For Comparison to

Kornhauser [18] Exp. Data: Run #12071539, 201.7 RPM,

1.008 MPa Mean Pressure

Figure 7: Modified CAST Calculations of Temperature Difference

from Heat Exchanger Center to Wall (34x20 grids, 120 time

steps/cycle). For Comparison with Kornhauser [18] Exp. Data:

Run #12071539, 201.7 RPM, 1.008 MPa Mean Pressure

CONCLUDING REMARKS

The Modified CAST compressible non-acoustic model agreed

well with 10 RPM gas spring hysteresis and P-V diagram data, and

also with compressible acoustic calculations of two-dimensional

velocities and temperatures. However, modified CAST calculations

deviated from experimental values of two-space (piston/cylinderheat exchanger) test rig data, although trends were predicted well.

Recent preliminary comparisons of the commercial CFDACE+ code compressible acoustic calculations with CAST, at CSU

(Ibrahim, et. al. [19]), for the two-space rig suggests temperature

agreement within about 2% for somewhat different grids. Therefore,

data and 2-D computations do not agree, for some reason, for the

two-space test rig. The reasons will be explored further. New data

will likely be required for further multi-D Stirling code validation

efforts.

When CAST and CFD-ACE+ calculations were made with the

same type of grid and time step size, CFD-ACE+ was about 8%

faster than CAST. Although not demonstrated here in comparisons

against the highly developed commercial CFD-ACE+ code,

Fedorchenko’s compressible non-acoustic technique, may have the

capability for reductions in simulation time for those transient

situations where compressibility must be simulated but acoustics

are not important.

REFERENCES

1. Ernst, W.D. and Shaltens, R.K., 1997, “Automotive Stirling

Engine Development Project,” DOE/NASA/0032-34, NASA

CR-190780, MTI Report 91TR15.

2. Dahr, M., 1999, “Stirling Space Engine Program,” Volume

1—Final Report, NASA CR 19999-209164/VOL1.

3. Tew, R.C. and Geng, S.M., 1992, “Overview of NASA

Supported Stirling Thermodynamic Loss Research,” NASA

TM–105690.

4. Thieme, L.G.; Qiu, S. and White, M.A., 2000, “Technology

Development for a Stirling Radioisotope Power System for

Deep Space Missions,” NASA/TM—2000-209767 (Prepared

for the 34th IECEC, Vancouver, B.C., August 1–5, 1999).

5. Furlong, R. and Shaltens, R., 2000, “Technology Assessment

of DOE’s 55-We Stirling Technology Demonstrator

Convertor (TDC),” NASA/TM—2000-210509.

6. Huang, S.C., 1993, “HFAST Version 2.0 Analysis Manual,”

Prepared under NASA Contract Number NAS3–25330

(NASA Glenn Research Center).

7. Gedeon, D., 1999, “Sage User’s Guide,” 3rd Edition, (Gedeon

Associates, 16922 South Canaan Road, Athens, OH 45701).

8. Geng, S.M. and Tew, R.C., 1992, “Comparison of GLIMPS and

HFAST Stirling Engine Code Predictions with Experimental

Data,” NASA TM–105549.

9. Tew, R.C. Jr., 2000, Two-Dimensional Compressible NonAcoustic Modeling of Stirling Machine Type Components,

Cleveland State University Doctoral Thesis.

10. Makhkamov, K. and Ingram, D.B., 2000, “Theoretical

Investigations on the Stirling Engine Working Process,”

Paper # AIAA–2000–2815, Proceedings of the 35th

Intersociety Energy Conversion Engineering Conference, Las

Vegas, NV, July 2000.

11. Fedorchenko, A.T., 1997, “A Model of Unsteady Subsonic

Flow with Acoustics Excluded,” J. Fluid Mechanics, Vol, 334,

pp. 135–155.

12. Peric, M. and Scheuerer, G., 1989, “CAST - A Finite Volume

Method for Predicting Two-Dimensional Flow and Heat

Transfer Phenomena,” GRS - Technische Notiz SRR–89–01.

13. Ferziger, J.H. and Peric, M., 1997, Computational Methods for

Fluid Dynamics, Springer-Verlag.

14. Patankar, S.V., 1980, Numerical Heat Transfer and Fluid

Flow, Hemisphere Ubl. Co., Washington.

15. Recktenwald, G.W., 1989, A Study of Heat Transfer Between

the Walls and Gas Inside the Cylinder of a Reciprocating

Compressor, University of Minnesota Ph. D. Thesis.

16. Kornhauser, A.A., and Smith, J.L. Jr., 1987, “A Comparison of

Cylinder Heat Transfer Expressions Based on Prediction of

Gas Spring Hysteresis Loss,” in Fluid Flow and Heat Transfer

in Reciprocating Machinery, T. Morel, J.E. Dudenhoefer, T.

Uzkan, and P.J. Singh (eds.), FED-Vol. 62, HTD-Vol. 93,

American Society of Mechanical Engineers, NY.

17. Kornhauser, A.A., and Smith, J.L. Jr. 1988, “Heat Transfer

During Compression and Expansion,” Phase II Progress

Report for Oak Ridge National Laboratory Subcontract

No. 19x-55915C, Cryogenic Engineering Laboratory,

Department of Mechanical Engineering, Massachusetts

Institute of Technology, Cambridge, MA.

18. Kornhauser, A.A., 1989, Gas-Wall Heat Transfer During

Compression and Expansion,” Massachusetts Institute of

Technology S. D. Thesis.

19. Ibrahim, M., Tew, R., Zhang, Z., Simon, T., and Gedeon, D.,

2001, “CFD Modeling of Free-Piston Stirling Engines,”

Proceedings of the 36th Intersociety Energy Conversion

Engineering Conference.
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Stirling cycle; Stirling engines; Simulation; Fluid mechanics; Heat transfer

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Subject Category: 34 Distribution: Nonstandard

Prepared for the 36th Intersociety Energy Conversion Engineering Conference cosponsored by the ASME, IEEE,

AIChE, ANS, SAE, and AIAA, Savannah, Georgia, July 29–August 2, 2001. Roy C. Tew, Jr., NASA Glenn Research

Center, and Mounir B. Ibrahim, Mechanical Engineering Department, Fenn College of Engineering, Cleveland State

University, 1960 East 24th Street, Cleveland, Ohio 44115–2425. Responsible person, Roy C. Tew, Jr., organization code

5490, 216–433–8471.

Roy C. Tew, Jr. and Mounir B. Ibrahim

Study of Two-Dimensional Compressible Non-Acoustic

Modeling of Stirling Machine Type Components

A two-dimensional (2-D) computer code was developed for modeling enclosed volumes of gas with oscillating boundaries, such as Stirling machine components. An existing 2-D incompressible flow computer code, CAST, was used as the

starting point for the project. CAST was modified to use the compressible non-acoustic Navier-Stokes equations to

model an enclosed volume including an oscillating piston. The devices modeled have low Mach numbers and are

sufficiently small that the time required for acoustics to propagate across them is negligible. Therefore, acoustics were

excluded to enable more time efficient computation. Background information about the project is presented. The

compressible non-acoustic flow assumptions are discussed. The governing equations used in the model are presented in

transport equation format. A brief description is given of the numerical methods used. Comparisons of code predictions

with experimental data are then discussed.