NASA Contractor Report 182108

,/

A Survey of Oscillating Flow in Stirling

Engine Heat Exchangers

(NASA-CR-182108) A SURVEY OF OSCILLATING

FLOg IN STIRLING ENGINE HEAT EXCHANGEHS

Annual Contractor Report [Ninnesota Univ.)

133 p CSCL 20D

G3/34

N88-22322

Unclas

014024_

Terrence W. Simon and Jorge R. Seume

University of Minnesota

Minneapolis, Minnesota

March 1988

Prepared for

Lewis Research Center

Under Grant NAG3-598

National Aeronautics and

SpaceAdministration

Table of Contents

Nomenclature ................................................ IiI

I •

1.1

1.2

INTRODUCTION

Purpose of this Report

Outline .................................................

•

2.1

2.2

2.3

SIMILARITY PARAMETERS

Isolating the Oscillating Flow Effect ................... 3

Physical Arguments for the Choice of Similarity

Parameters .............................................. 8

Derlvatlon of Simllarlty Parameters ..................... 12

•

3.1

3.2

3.3

3.4

3.5

3.6

3.7

FLUID MECHANICS OF OSCILLATING PIPE FLOW ................. 16

Analysis of Lamlnar Flow in a Straight Pipe ............. ]8

Laminar Oscillating Flow in Curved Pipes ................ 25

Non-slnusoldal and Free Oscillations .................... 27

Transition from Laminar to Turbulent Flow ............... P8

Turbulent Flow .......................................... 35

Entrance and Exit Losses ................................ 37

Compressibility Effects ................................. 38

4. HEAT TRANSFER IN OSCILLATING PIPE FLOW ................... 4]

4. I Qualitative Considerations .......................... . . . .42

4.2 Axial Heat Transfer in Laminar Oscillating Flow ......... 46

4.3 Experimental Data ....................................... 48

•

5.1

5.2

5.3

FLUID MECHANICS AND HEAT TRANSFER IN REGENERATORS

Steady Flow ............................................. 54

Unsteady Flow and Heat Transfer In Regenerators ......... 60

Regenerator Theory ...................................... 61

•

6.1

6.2

STIRLING ENGINE DATA BASE

Evaluation of Similarity Parameters ..................... 63

Documented Engines ...................................... 65

•

7.1

7.2

7.3

RESULTS AND DISCUSSION

Heaters and Coolers ..................................... 68

Regenerators ............................................ 77

Similarity Parameters ................................... 80

8. CONCLUSIONS AND RECOMMENDATIONS .......................... 82

ACKNOWLEDGEMENTS ............................................. 84

REFERENCES ................................................... 85
APPENDICES

A A SURVEYOF FRESSURE, REYNOLDS-NUMBERAND MACH-NUMBER

VARIATIONS............................................... 93

B DERIVATION OF SIMILARITY PARAMETERS...................... ]08

C VELOCITY PROFILES IN LAMINAR FLOW........................ ]]5

D EQUATIONSOF OBSERVATIONSOF TRANSITION.................. ]]8

E EFFECTS OF PRESSUREPROPAGATION.......................... I]9

F AUGMENTATIONOF AXIAL TRANSPORT.......................... ]25

lJ
Nomenclature

symbol

a

A R - Xmax/£

Au

Cp

d

dh

D

De

Ec

f

f

h

void volume

surface area

-Re 

Um _ m_ 2

wm

Cp (T H - T L )

= (-dp/dx)

p V _

I¢

k

k

1

£

£

£h

L

L

m

M - u/a

Mmax

Nu -

P

Pr -

r

Rc

Re -

- Um,max/a

hl/k

Cpu/k

u L

units

m/sec

J/(kgK)

m

m

m

set-1

W

W

Wl(mK)

m2

m

m

m

m

m

m

kg

Pa

W

m

m

explanation

speed of sound

relative amplitude of fluid

displacement

augmentation coefficient

specific heat at constant pressure

diameter

hydraulic diameter

pipe diameter

Dean number

Eckert number

frequency

friction factor

heat transfer coefficient

integrated dissipation function

thermal conductivity

permeability

length

heat exchanger length

connecting rod length

heated length of heat

characteristic length

engine flow length

mass

Mach number

Mach number based

amplitude

Nusselt number

pressure

Prandtl number

on the

exchanger

velocity

heat transfer rate

radius

pipe radius of curvature

Reynolds number based on length L
Remax

Rem

Str -

t

t o

TiJ

U

U

U

V

VT

X

X

Xmax

Y

Um,max

see

see

Pa

m/see

m/see

m/see

m/see

m

m

m

m

Reynolds number based on the

amplitude of the cross-sectional

mean velocity

dimensionless frquency,

Valensl number,

kinetic Reynolds number

Strouhal number

time

characteristic

stress tensor

t ime

velocity

streamwise velocity

characteristic velocity

superficial velocity

dimensionless tidal volume

distance, location

streamwlse coordinate

amplitude of fluid displacement

cross-stream coordinate

Greek

P

¢

¢

¥

- 2wf

o tad

m

N see

m_rm2/sec

kg/m3

Pa

W/m3

rad/sec

augmentation factor

symbol of proportionality

ratio of specific heats

coefficient of excess work

lead angle

wave length

dynamic viscosity

kinematic viscosity

density

shear stress

porosity

dissipation function

axial augmentation function

angular velocity

Superscripts

#

normalized quantity

at shock formation

iv
Subscripts

C

h

H

L

Im

m

max

0

P

r

W

T

cold end

hot end

high temperature

low temperature

log-mean

average over cross-sectlon

maximum during one cycle

reference state

of pressure

of the regenerator

at the wall

of wall shear stress

I. INTRODUCTION

1.I Purpose of thls Report

Two design objectives for Stlrllng engines are to keep mechanical

losses low and to achieve good heat transfer from and to the working fluid.

These objectives conflict because heat exchangers with higher heat transfer

effectiveness generally have higher flow losses. It Is therefore valuable

for the designer to know the trade-off between pressure drop and heat

transfer. Thls Is particularly important when hlgh performance Is required

as _r the Automotive Stirllng Engine and the Space Power Demonstrator

Engine.

Computer modelling is used to evaluate options in Stlrllng engine

design. Measured Stlrling engine efficlencles and power output are often

lower than predlctlons would indicate, however. Transient pressure

variations in the Stlrllng engine working spaces are also often not

predicted correctly (Rix 1984). Inadequate modelling of fluid mechanlcs and

heat transfer in the heat exchangers (usually by using steady-state

correlations for pressure drop and heat transfer) may partially account for

thls discrepancy. Improving thls model will require more physically correct

expressions for the effect of oscillation. It is hoped that thls report

will make a step In that direction by improving the understanding of fluid

mechanics and heat transfer In Stlrllng engines. It summarizes, extends and

discusses research found In the literature and draws tentative conclusions.

1.2 Outline

The oscillating flow effect Is isolated as the focus of thls report.

Similarity parameters to characterize fluid mechanics and heat transfer In
Stirllng engine heat exchangers are proposed. A literature review and some

analyses address fluid mechanics and heat transfer in oscillating flow in

pipes under conditions which are similar to conditions in heaters and

coolers. Literature which deals with flow in porous media, similar to

regenerators, Is also reviewed. The operating characteristics of 11

Stlrling engines are described in terms of similarity parameters and

tentative conclusions are drawn about the fluid mechanics and heat transfer

conditions in Stirling engine heat exchangers. Open questions to be

answered by future research are identified. Some of the material covered In

this report is summarized in Seume and Simon (1986).
2. SIMILARITY PARAMETERS

Similarity parameters are required to concisely and generally describe

fluid mechanics and heat transfer in physically similar systems. Their

proper choice Is crucial for the development of an experimental program and

the interpretation of experimental data.

Section 2.1 discusses some characteristics of the flow in Stlrling

engine heat exchangers that are not described by similarity parameters--in

particular, slmpllfylng assumptions to isolate the oscillating flow effect.

Section 2.2 presents physical arguments for the chosen set of similarity

parameters and Section 2.3 derives the similarity parameters by normalizing

the governing equations.

2.1 Isolatln_ the Oscillating Flow Effect

The effect of oscillation on fluid mechanics and heat transfer has been

the subject of several studies related to Stirling engines (e.g., Kim 1970,

Organ 1975, Miyabe et al. 1982, Chen and Griffin 1983, Hwang and Dybbs 1980

and 1983, Taylor and Aghili 1984, DiJkstra 1984, Jones 1985, Rice et al.

1985). In review of these studies, it became clear that the effect of

oscillation on pressure drop, viscous dissipation and heat transfer must

first be studied isolated from the remainder of the processes occurring in

the engine, in particular isolated from:

- compression and expansion of the working fluid

- non-harmonlc fluid motion

- high temperature gradients.

Some discussion about each of these effects in the Stirllng engine follows.
Compression and expansion of the working fluid. The working fluid in

Stirllng engines moves from the compression space toward the expansion space

and back during each cycle. Simultaneously, it undergoes compression and

expansion (Figure 2-I). Therefore, the mass of fluid in the heater in

Figure 2-I varies with time and consequently the mass flow rates at the ends

of the heater differ throughout the cycle (full and dashed line).

Appendix A provides a survey of the cycle variations of pressure and

Reynolds numbers in Stlrling engine heat exchangers using the isothermal and

Schmidt analyses (c.f. Section 6.1). The engines and their operating points

are described in Section 6.2. Figure 2-I shows that the pressure change is

fastest when the fluid velocity In the heater is near zero. In the cooler

and in the regenerator, however, velocities are high when the pressure is

changing most rapidly.

During compression, the gas temperature rises. This decreases the

temperature difference between the bulk of the gas and the heater wall and

thereby may reduce the heat transfer from the heater pipe to the gas. In

the cooler, compression increases the temperature difference between the

bulk fluid temperature and the wall temperature. The heat transfer may,

however, be out of phase with the bulk-to-wall temperature difference. This

was shown by Faulkner and Smith (1984) in a study related to heat transfer

in the cylinders of reciprocating machinery. In the heater, fluid

velocities are low during expansion and compression. Consequently,

convective heat transfer is not expected to be as strong as in the cooler

and regenerator during compression and expansion.

In the cooler and regenerator, temperature in the gas increases

simultaneously with convective heat transfer during the cold blow, i.e.,

when gas moves from the compression space towards the expansion space.

4
--

\

\

\

% i

I

I

I

I

I

I

I

I

I

/ \

\

|

_00

I

io+o

I. I

II1,0+11 JPlO,O

CFII_NK FIN6LF

I

Figure 2-1: Variation o f pressure and Reynolds number

in the heater of a GPU 3, isothermal analysis.

Flow toward the cold end is poslti_;

full llne = hot end, dashed line = cold end.
During the hot blow, the gas expands as it flows through the cooler and

regenerator.

DiJkstra (1984, p. 1886) proposed to model the compression of the

working gas as bulk heating of a fluid flowing through heat exchangers. He

pointed out that the convective heat transfer coefficient in the case of

bulk heating is very different from that for specified wall temperature or

specified heat flux. The differences are particularly blg in the entrance

region of tubes and in the turbulent flow regime. It is, however, not clear

whether bulk heating can be used to model temperature changes in the working

gas due to compression and expansion.

This report focuses on the effect of the oscillation of the working

fluid on fluid mechanics and heat transfer In the heat exchangers. To

isolate this effect, compression and expansion of the working fluid are

neglected in most of the discussion below.

Mean velocity variation. In Figure 2-2, the mean velocity is

normalized with the speed of sound to form a Mach number; according to the

assumptions implicit in the isothermal analysis, the speed of sound is

uniform and constant in each heat exchanger. Figure 2-2 shows that the mean

velocity variation in a typical Stlrllng engine is roughly sinusoidal and

very similar to the Reynolds number variation shown in Figure 2-I.

Velocity and Reynolds number variation deviate from a sine function in

that:

(i)

(2)

the hot blow period, during which the flow is positive (i.e.,

towards the compression space), is shorter than the cold blow

period.

The maximum velocity during the hot blow period is greater than

during the cold blow period.
°;

! I I i

_0.0 O0.O llkO,4 P40.O IO, O

CP.gNII RNGLE

I

_.0

Figure 2-2: . Variation of Mach number in the heater

of a GPU 3, isothermal analysis.

Flow toward the cold end is positive;

full line = hot end, dashed line fficold end.
The differences of the mass flow variation (Figure 2-I) and the mean

velocity variation (Figure 2-2) from harmonic motion will be neglected in

the discussion below.

Temperature dependent properties. The molecular viscosity and thermal

conductivity of gases increase with temperature. Due to the temperature

gradients in the heat exchangers, the gas properties vary in the radial and

axial directions.

In particular, the viscosity near the heater wall is expected to be

higher than in the core; the viscosity of the gas is lower near the wall of

the cooler than in the core of the flow. Therefore calculations based on

the assumption of constant properties in this report should only be

considered approximate. Some authors consider density variations (buoyancy

effects) in turbulent forced convection with large density gradients

important. For a review of this subject, see Petukhov et al. (1982).

2.2 Physical Arguments for the Choice of Similarity Parameters

This section describes qualitatively some of the transient phenomena

expected in Stlrllng engine heat exchangers. Similarity parameters

characterizing each of these phenomena are proposed.

Velocity and acceleration. In steady flow, the friction factor (a

non-dlmenslonal viscous pressure drop) is a function of Reynolds number (a

non-dlmensional mass flux). Oscillating flow experiments, e.g. those by

Taylor and Aghill (1984) showed that the functional relationship between

friction factor and Reynolds number is different for oscillating flow than

for steady unidirectional flow.

In the limit of very slow oscillations, the oscillating flow

relationships should approach that of steady flow. This suggests that the

8
friction factor in oscillating flow is also a function of a dimensionless

frequency or dimensionless acceleration. Following Chenand Griffin (1983),

this dimensionless frequency is chosen as:

d =

Re =

4 v

In this report, the Reynolds number is based on the maximum velocity in the

cycle:

Um,m_xd

Remax = v

Fluid displacement. Walker (1962, 1963 and 1980, p. 130) shows that

the fluid displacement in Stirling engines may be so small that some working

fluid merely moves back and forth within the heat exchanger without exiting.

If the gas moves as a plug and if there is no axial mixing, this gas

contributes to the axial heating only by absorbing thermal energy from the

wall or adjacent fluid at the high temperature end of the heat exchanger and

releasing it at the low temperature end. Only the gas that enters and

leaves the heat exchanger transfers heat directly to the volume in the

cylinders. Organ (1975) argued that the gas that stays within the heat

exchanger may be considered an undesirable addition to the dead space in the

engine. The amplitude ratio, AR, is used to describe fluid displacement in

heat exchangers. It is the fluid displacement during half a cycle divided

by the tube length, computed by assuming that the fluid moves as a plug at a

mean velocity, um. This is the reciprocal of ^ introduced by Organ (1975).

If AR • I, most fluid oscillates within the tube without exiting; if AR _ I,

the fluid traverses quickly through the tube, residing in the upstream and

downstream spaces during most of the cycle.
Entrance and exit effects. The pressure drop at the entrance of a heat

exchanger has two components. The reversible pressure drop is the result of

an increase in kinetic energy by expending flow work (pressure drop) in the

contraction upon entrance to a heat exchanger. In the case of a

compressible working gas, density will decrease as the gas is expanded

(approximately adiabatically) in this contraction. The irreversible part of

the entrance pressure drop is due to viscous dissipation in the shear flow

of the vena contracta. It is typically small in steady flows. At the exit,

pressure may be recovered as the gas velocity decreases. The pressure rises

and kinetic energy is lost irreversibly by viscous dissipation in the shear

layers of the separated flow upon exit from the heat exchanger into the

downstream chamber.

Since the magnitude of entrance and exit losses depends on the area

ratios of the contraction and expansion of the duct, these area ratios (or

the corresponding diameter ratios, (d/D)) are added to the Reynolds number

as similarity parameters for capturing entrance and exit losses.

Pipe curvature. The pipes in Stifling engine heaters are often curved.

It is known from steady unidirectional flow that secondary flows develop in

curved pipes as centrifugal forces act on the fluid. The Dean number,

De = Re 2/_ c ,

where Rc is the radius of curvature of the pipe centerllne, is commonly used

as a similarity parameter for this effect (Berger et al. 1983).

Developing flow. In steady flow, the velocity profile in a pipe

changes in the flow direction until fully developed conditions are reached.

The entrance length scales on the diameter of the pipe (or the hydraulic

diameter of the duct). The friction factor in the entrance (developing)

10
region is higher than in the fully developed region. Therefore, the average

friction factor for developing duct flow is a function of the

length/diameter ratio, _/d, as well as the Reynolds number.

Flow in regenerators. Flow in the complex geometries of Stlrling

engine regenerators is commonly described in terms of a Reynolds number

based on the hydraulic diameter and the matrix porosity. Other choices of

similarity parameters are discussed in Section 5.1.

Compressibility of the working gas. The compressibility of the working

gas affects the entrance and exit losses of the heat exchangers as described

above and the propagation of pressure changes throughout the engine. If the

gas velocity in a heat exchanger is of the same order as the speed at which

pressure changes propagate (approx. the speed of sound), pressure (and,

therefore, density) variations in the engine due to compressibility will be

significant. As the ratio of gas velocity to speed of sound (i.e. the Mach

number) approaches unity, shock waves form and the flow becomes choked. The

Mach number, based upon the highest velocity in the engine cycle, is

therefore chosen as a measure of compressibility effects.

Temperature transients. The walls of the heater and cooler and the

matrix of the regenerator undergo temperature transients as they are

subjected to different gas temperatures during hot and cold blows. The

transient response, particularly of the regenerator matrix, affects the

engine performance (Walker 1980, p. 140). This response is described by

dimensionless parameters from regenerator theory, e.g. dimensionless

regenerator length and dimensionless blow period, as discussed in Section

5.3. Also, the transient response of the heater and cooler walls may

contribute to enhanced axial heat transfer as discussed in Section 4.1.

]I
2.3 Derivation of Similarity Parameters

Momentum equation. Similarity parameters, introduced in Section 2.1,

can be derived by normalizing the momentum and the energy equations.

Neglecting gravity, the momentum (Navier-Stokes) equation is:

"_ "_ Vp "_

au + u.vu - + vv2u

at p

A normalized form suitable for pipe flow is derived in Appendix B:

_'_+ 2 = - 2 p

Here the superscript * denotes a normalized quantity. The dimensionless

frequency, Rem, is the coefficient of the temporal acceleration (or

unsteady) term, whereas the maximum Reynolds number, Rema x, is the

coefficient of the spatial acceleration and the pressure gradient term.

The viscous term has no coefficient. Thl8 does not imply, however,

that Rema x is the ratio of the _.V_ (steady inertia) term to the v?2_

(viscous) term. That would only be true if both non-dlmenslonal terms were

of the same order, i.e., if

-

This is not necessarily the case. In a fully developed laminar pipe flow,

for example, the _.V_ term vanishes but the Reynolds number is still

non-zero.

The dimensionless frequency, Re_, has also been called the kinetic

Reynolds number (White 1974, p. 144) or the Valensi number (Park and Baird

12
1970). It Is a multlple of the Stokes number (Grassmann and Tuma 1979) and

It Is the square of the Womersley parameter.

A slightly rearranged version of the normallzed momentum equation Is:

V* N Str P.v "- - ÷ 2

2 atT + p Rema x

where Str- _d/um,ma X is the Strouhal number. Note that the definition of

the Strouhal number may vary in the literature (Telionls 1981, p. 88), and

that Str is a multlple of the Strouhal number used by DiJkstra (1984).

Geometric slmllarlty. The oholoe of length scales for the

normalization of the momentum equation is arbitrary. To maintain

similarity, however, all dimensions must scale on the same length; in this

case the pipe radius or the diameter d was chosen. Therefore, (£/d) is the

appropriate similarity variable describing the pipe length. The relative

amplitude of fluid displacement is not an independent similarity parameter.

For slnusoidal fluid motion, It is shown in Appendix B to be:

AR " 2 d Re

Description of geometries other than straight pipes, such as the

various regenerator types, curved pipes, or ducts of rectangular

cross-section, require additional descriptors to complement the similarity

parameters described in this section, e.g. the Dean number.

Energy equation. Heat transfer is governed by the energy equation:

pep (_-_ _T + _-gT) - Bt_P + _'Vp + V.(kVT) +

]3
where ¢ is the dissipation function (Appendix B). A normalized form for

pipe flow is:

, . aT* _" V"T*)

p Cp (2 Rem -_ + Remax "

= 2 Re Ec ap*

_ + Ec Rema x _*.V*p* +

where Pr -

ko

EC = Um'm_Y_

Cpo (Th-Tc)

V*-(k*V*T*)

Pr

+ Ec ¢*

Other choices of similarity variables. Other forms of the normalized

governing equations are obtained if the variables are normalized differently.

If the temperature is normalized as T* - T/T o and we assume that the fluid

is an ideal gas with constant specific heats, the Eckert number, Ec, can be

replaced by (Y - 1) Mmax 2

The Mach number can also enter the normalized momentum equation through

the equation of state if pressure is normalized with a reference pressure:

p* = 2_ Po = PoRTo = ._my

Po r'_ax

Then the Math number appears in the pressure term:

Re a_* R--£max _t"V*_* Re_.£max I _ + v*?*'_*

"_ + 2 " - 2 YMma X 2 p

These examples show that the choice of similarity parameters is

somewhat arbitrary. The appearance of a dimensionless parameter in a

normalized equation alone does not imply that it is a relevant similarity

parameter. Meaningful similarity parameters are obtained when all

normalized terms in an equation are of order I, e.g. in the momentum

equation:

14
._u*. ,@u*.___} I d.p__) . O(v* _2u*) 0 (_-_1 - o(u - 0(3.dx - I

Then the coefficients of those normalized terms, i.e., Re_, Rema x, Str in

the momentum equation, represent the relative magnitude of the terms in the

dimensional momentum equation, e.g. Rem is the ratio of the temporal

acceleration term to the viscous term of the momentum equation in a laminar

oscillating pipe flow or the Dean number is the ratio of the spatial

acceleration term (due to centrifugal force) to the viscous term of the

momentum equation in a laminar flow in a curved pipe.

In the energy equation, an Eckert number on the order of I would

indicate that viscous heating is important and a Math number of order I

would indicate that density variations due to high fluid velocities are

expected.

15
3. FLUIDMECHANICSOFOSCILLATINGPIPE FLOW

In unsteady flows, pressure and shear forces in the fluid are not in

equilibrium; instead there is a local balance of inertia, pressure and shear

forces. Therefore, the fluid mechanics of oscillating flows are different

from those of steady flow.

Figure 3-I presents an integral momentum balance on a cylindrical

control volume in fully developed flow (u-Vu - 0). The oscillating flow

case shows the additional term associated with temporal acceleration (_u/_t,

where u is the cross-sectlonal mean velocity).

16
_D2p(x) ()

L

steady flow '-

T

W

_X

;___D2p( x+Ax

(a) pressure In balance with wall shear stress

T

W

oseillatin_ flow

(b) pressure In balance with wall shear stress + fluid inertia

Figure 3-I: Force balance in fully developed plpe flow.

17
3.1 Analysis of Laminar Flow in a Straight Pipe

Measurements by Richardson and Tyler (1929) first indicated an

oscillating flow effect. They found a maximum velocity near the wall for

oscillating pipe flow. Sexl (1930), Womersley (1955) and Uchlda (1956) have

since shown thls by analysis of both sinusoldal and non-slnusoldal

oscillating flows. Similar analyses were performed by Drake (1965) for

rectangular channels and by Gedeon (1986) for flow between parallel plates.

Zielke (1968) and Trlkha (1975) used Laplace transforms to calculate

pressure drop in unsteady laminar flow.

Figure 3-2, based upon the Uchida (1956) analysis, shows velocity

distributions for fully-developed flow at several times during the cycle and

for several values of Re_. All profiles assume the same magnitude of

pressure gradient oscillation, and the local velocities are normalized with

the maximum mean velocity that would occur with steady laminar flow

(responding to the cycle-maxlmum pressure gradient). Velocity distributions

similar to these were measured by Shlzgal et al. (1965, p. 99) and Edwards

and Wllkenson (1971, p. 87). An analysis similar to Uchlda's but for flow

between parallel flat plates was made by Gedeon (1986). Figure 3-2 shows

parabolic distributions for slowly oscillating pipe flow (Rem - 1). With

higher Rem, however, the velocity amplitude decreases and, during parts of

the cycle, the near-wall flow direction is opposite to the mean flow

direction (Rem - 30). A velocity maximum develops near the wall. At the

maximum frequency (Re_ - 1000), this free shear layer becomes narrower and

moves closer to the wall. A unlfo_m velocity core exists and velocity

gradients become concentrated in a Stokes layer near the wall.

]8
0

f

0

0

I

0

l _ a _ g

I

O 0 O 0 O

d d 9 9

z+O I. X AJ.IOO73A C]=IZI7VI_I::ION

o _

8

I I I I

,_. o d _. _'.

AJ.IOO7:iA CFIZI7VI_II:::ION

O O O O • © c_ _,

d o 'T

AIIOO73A 03ZI7V_IdON

Ig

CO

m

C_

<

_. n"

o U.I 0

_ o

0_ 0

o

0_ II

- 8

9 _

0

0

..J

"T

0 _

•- if)

a

in tr

° W

_. L II

8

o _j

0

0

..J

0 v

9

a

<

o t,U

n

n

_. _ It

8

rr

<

0

0

._1

o _

o

u_

-H

.11

o

I--I

u_

_2

1.1

,.-I

_J

o

o

-1-1

I

_o

,1-1
The presence of a large velocity gradient between the maximum velocity and

the core (the free shear layer) may play an important role in

laminar-to-turbulent transition. Further velocity profiles are shown in

Appendix C.

To discuss the effect of flow oscillation on shear stress and pressure

drop, augmentation factors, o r and ap, and lead angles, Ar and Ap, are

introduced as:

um - Um,ma x cos (rot)

_w _ or 8_/d Um,ma x cos (mt + AT)

Ap - ap 32pl/d 2 Um,ma x cos (mt + Ap)

Note that for steady flow, m r and ap are unity and Ar and Ap are zero.

These quantities, taken from the Uchlda (1956) analysis, are plotted versus

Rem on Figures 3-3 and 3-4. Note (Fig. 3-3) that for the Re m range

chosen, the wall shear stress is enhanced by a factor of 8 over that of

unidirectional flow, and that the pressure change is enhanced by a factor of

130.

The phase relationships given in Figure 3-4 indicate that, in the limit

of high Rem,the shear stress leads the mean velocity by 45 ° and the

pressure change leads the mean velocity by 90 °. The near-wall fluid

responds more quickly to the imposed pressure gradient than does the higher

inertia core flow. Clearly, neither the shear stress nor the pressure drop

should be computed from the pressure gradient by assuming quasi-steady flow.

In oscillating flow, loss of engine work due to viscous dissipation

cannot be calculated by multiplying mean velocity and wall shear stress as

is done in steady flow. Instead, the viscous power dissipation in a pipe

section of length _ must be computed as:

2O
2

I0

I0

•Augmentation of pressure change, (Zp

Augmentation of wall shear stress, o_

I0 Re=, 102 103

Figure 3-3: Coefflclents of amplitude of pressure drop

and wall shear stress

21
90 ° , , I I I , , ,I , I , ! , I , ,j , , , | , , ..]

t

_°°" _ -!

X '_T

I0 I0 103

Re_

Figure 3-4: Lead angles of pressure drop and wall shear stress

22
I¢ " 2_ & r-od/2 tJ(Bu)= _ r dr

The relative increases (due to flow oscillation) of shear (irreversible)

work and total pumping work (reversible and irreversible) over those for

steady flow can be expressed by the coefficients of excess work, eI and Cp,

respectively.

c I -

[;o27 I¢(mt) d(wt)]

[;o27 I_(wt) d(wt)]Re +0

2_ [;o

¢P " 27 [; o - (_x)Um

d(_t) ]

d(wt) ]Re ÷0

These quantities are plotted versus Re_ on Figure 3-5. Note that over the

Re= range shown, the'irreversible work increases by a factor of six,

whereas the pumping work increases by a factor of 125. The difference

between the pumping and shear work values represents reversible work. In

the Stirllng engine most of this is eventually lost due to dissipation upon

entry to or exit from the heat exchanger channels. Some of this reversible

work may be recovered as gas exiting the channels impinges upon and does

work on a piston surface.

23
103 ' i ' ' ' L

2

I0

£

P

C i

IO

£p,reversible + irreversible

£i ,irreversible

I0 10 2 103

Figure 3-5: Relative increase of pumping work due to flow oscillation

24
3.2 Laminar Oscillating Flow in Curved Pipes

The pipes in many Stlrling engine heaters are bent. If the radius of

curvature of a pipe is small, secondary flows are strong. This is true for

steady and for oscillating flows. Due to the pipe curvature, centrifugal

forces act on the fluid; therefore the spatial acceleration term u. Vu is

non-zero while it is zero in fully developed laminar flow in a straight

pipe.

In the normalized momentum equation, the Dean number replaces the

Reynolds number. The unsteadiness is again characterized by Rew.

Telionls (1981, p. 183) gives a different but equivalent set of similarity

parameters. Figure 3-6 shows a schematic of oscillating flow in a curved

pipe. In each half of the pipe there are two counter-rotatlng secondary

flows: one in the potential core and one in the Stokes layer.

25
\

a - d/2 ro - Rc

Figure 3-6: Oscillating flow in a curved pipe

(from Tellonis 1981, p. 182)

Yamane et al. (1985) calculated these streamlines and axial velocity

profiles. The latter were confirmed experimentally by Sudou et al. (1985).

Streamlines and velocity profiles for laminar, oscillating flow in

rectangular curved ducts were calculated by Sumida and Sudou (1985).

Yamane et al. (1985) calculated and measured the increase of wall shear

stress and pressure drop in a curved pipe over that in a straight pipe.

They presented these ratios as a function of Dean number and Womersley

parameter.

26
3.3 Non-sinusoldal and Free Oscillations

Section 3.1 discussed sinusoldal oscillations in a straight pipe.

Uchida (1956) used a Fourier series representation to extend this analysis

to general, non-sinusoidal periodic laminar flows. This analysis may be

useful since oscillations in Stirllng engines are generally not sinusoldal.

Chan and Baird (1974) studied forced oscillations of liquid columns in a

range that is of interest for the design of liquid piston engines (West

1983, pp. 54-59, 131). Free (damped) oscillations of liquid columns in

U-tubes were studied by Valensi (1947), Park and Baird (1970), and Iguchi et

al. (1982). It is not clear to what extent their results regarding fluid

friction and the transition from laminar to turbulent flow (see Section 3.4)

can be applied to the forced oscillations of liquid columns occurring in

liquid piston engines.

The effect of pipe curvature on the flow was neglected in thestudles

referred to above and in the transition studies discussed in the next

section. The authors assumed that the curvature effect is negligible

because the pipe radius of curvature is small compared to the pipe radius.

27
3.4 Transltlon from Laminar to Turbulent Flow

The transltlon process. Transltlon In unldlrectlonal steady flow Is

known to be sensltlve to bulk mean veloclty and acceleratlon. Lower

velocltles and acceleration stab111ze, whereas higher velocltles and

deceleration destablllze. Since In osclllatlng flows both veloclty and

acceleration vary, It Is expected that flow patterns may change from

lamlnar-llke to turbulent-llke throughout the cycle under near-crltlcal

conditions. Hlno et al. (1976) probed an osclllatlng plpe flow wlth a

single hot-wlre and presented the traces of absolute value of the velocity

shown in Figure 3-7. The parameter n is the radial locatlon wlth n " 0

being the center of the plpe. The traces show a lamlnar-llke flow during

acceleration and a turbulent-like flow durlng deceleration. Wlth Increased

Rema x (5830, Flg. 3-7b), the turbulence perslsts into the acceleration

portlon of the cycle. From these traces, one would expect transltlon to

extend over a broad range of Rema x. Thls supposltlon Is conslstent wlth

measurements taken by Ohm1 et al. (1982) where a wlde range in Rema x was

observed between lamlnar-llke and turbulent-like osclllatlng flows. The

slgnal in the first part of the cycle shows slightly stronger fluctuations

than that In the second part, probably because of flow separation In the

asymmetric apparatus. At the same dlmensionless frequency (Re m - 7.30)

but a higher amplitude Reynolds number (Rema x = 5830) the turbulent

fluctuations are stronger than at Rema x - 2070 (Flgure 3-7(a); they start

prlor to deceleratlon durlng the hlgh velocity part of the cycle.

28
_t-O

.j _ _, centerline

4

0

=l 0 L.

el - _ F near.wall I

, _ J,,

(o) Rew= 7.30 Remo=x2070

i

O"

2.

centerline

1. near wall

0

, , _, ;, 0 ;. •

==¢

(b) Re== 7.30 Remo-'x5830

e/= local radius / pipe radius

Figure 3-7: Velocity measur_ents taken with a hot wire

by Hino et ai.(1976)

29
The ensemble-averaged velocity is lower than the laminar velocity at

the center of the pipe and higher than the laminar velocity near the pipe

wall. This indicates that the profile becomes flatter, which is

characteristic of turbulent velocity profiles. This observation is

consistent with the work of Ohmi et al. (1982) who found that the velocity

profile during the laminar part of the cycle agrees well with the

theoretical laminar solution (see Section 3.1) and that the velocity profile

during the turbulent part agrees well with the I/7 power law for steady

turbulent plpe flow (see e.g. Schllchtlng 1979, p. 597-602).

Experimental observations of transition. Figure 3-8 shows observations

of transition. Ohmi et al. (1982) studied forced oscillations of a gas in a

straight pipe. They did not state their criterion for transition. Iguchi

et al. (1982) observed free oscillations of a liquid column in a U-tube.

They chose the laminar-to-transitional flow line to be where the amplitude

of oscillation began to deviate from that of the laminar solution discussed

earlier. The transitional-to-turbulent flow llne is where measured

amplitudes of oscillation began to agree with predictions based on

I/Tth-power turbulent plpe flow profiles. Park and Baird (1970) observed

the decay of free oscillations of liquid in a manometer. They calculated

cycle maximum wall shear stress from the observed amplitudes of oscillation

using two methods. The first was based on a laminar flow prediction, the

second on the turbulent I/Tth-power profile. They assumed transition to

occur at the amplitude where the maximum wall shear stress, calculated from

the laminar prediction, exceeded that calculated from the I/7th-power

profile. They attribute their data scatter to column end effects in the

liquid column which are a function of L2_/_ where L is the length of the

3O
10 6 _

I05 _

LJ

I0 4 _

.

10:3

I

' ' I ''''1

Fig 5(b) +

Fig 5(a)+

' ' ' I .... I

1 ! • 1 1TT,

Park and Baird(

Iguchi et al.(19ez

;ergeev(t966)

m

m

m

Ohmi et o

Grassmann and Tu

(1979)

! I I I

10 3

Re_,

Figure 3-8: Observations of transition in oscillating flow

31
liquid column. These end effects may be important in liquid piston Stirling

engines. Sergeev (1966) did not state his criterion, but, since he used

aluminum particle flow visualization, we surmise that transition was

observed through the transparent walls as a change in the flow structure.

His work was in stralght-tube, forced-oscillatlon flow. Grassmann and Tuma

(1979) used an electrolytic technique to observe turbulent fluctuations In

wall mass transfer rate, thus locating transitional behavior.

Merkli and Thomann (1975) studied the transition from laminar to

turbulent oscillating flow at dimensionless frequencies (Re_ -

2500...4000) beyond the range presented in Figure 3-8. Their apparatus was

a piston oscillating in a resonance tube. They noticed a secondary weak

vortex motion outside the oscillating boundary layer. The tube was operated

near resonance. The secondary vortex observed by Merkll and Thomann may be

related to the vortex street that Sobey (1985) observed in steady and

oscillating channel flow. Sobey also predicted the vortex street

numerically and considered it a result of shear-layer instability.

DiJkstra (1984) observed, in the transitional regime, that water which

was initially in the tube would remain laminar, while water entering the

tube would flow in a turbulent pattern and would remain turbulent in the

tube.

Though the transition predictions differ with criteria, the researchers

agree that transition Rema x increases with Re_. The sequence of the

transition predictions is consistent with the criteria used. Iguchi's et

al. (1982) lower line is based on the first sign of deviation from laminar

behavior. The Grassmann and Tuma (1979) criterion is based on fluctuations

at the pipe surface. These fluctuations are likely to occur at about the

same Rem as that established by Sergeev (1966) based on the onset of

32
turbulent motion of particles. Finally, the Iguchi et al. (1982) upper llne

is based _n agreement with the predictions from the I/7th-power law. It

agrees with the observations by Ohml (1982) In forced oscillation of gas in

a straight pipe. Equations for these observations of transition are listed

in Appendix D.

Theoretical prediction of transition. Von Kerczek and Davis (1972)

used the energy method to predict a lower bound for the instability of

Stokes layers on a flat plate. Figure 3-9 shows their results in the form

of the critical value of the Reynolds number Rema x below which the

oscillating flow cannot go unstable. This lower bound underpredicts the

transition Reynolds number by roughly one order of magnitude. The trends of

the two curves, however, agree well.

Cayzac et al. (1985) presented predictions for the lower bound of

stability In oscillating pipe flow. Like Von Kerczek and Davis, they found

good qualitative agreement in the trends but a quantitative discrepancy of

one order of magnitude. For steady pipe flows, they predicted the lower

bound for instabilities to be Re D _ 750 while Re D - 2000 is the well-known

experimental value.

The theoretical prediction of the lower bound of instability by the

energy method apparently does not yield results that are practically useful.

For other theoretical approaches to the stability of oscillating flows see

Davis (1976).

33
I0- - i ' i ' '''i , , ' i ' ' ''i ' ' ' I , i,,.

- !

I

R%,= ! _ _ __

I0

i

Van Kerczek 8 Davis (19

I I , I , illl I I I I J llll = , , I , ,,,i

I I0 10 2 10 3

Figure 3-9: Theoretical prediction of transition

and experimental observations

34
3.5 Turbulent Flow

Though much remains to be learned about the effect of oscillation on

turbulent flows, significant findings have been documented.

Quasi-steady approach. DiJkstra (1984) and Vasiliev and Kvon (1971)

applied turbulence models taken from steady flow to oscillating and

pulsating pipe flow, respectively. Kirmse (1979) compared Vasiliev and

Kvon's model to his experimental data and concluded that their quasi-steady

prediction was not adequate.

The concept of an oscillation-sensitive and a quasl-steady turbulent

regime was introduced by Ohmi et al. (1982, p. 536). In their experiment,

the crank radius of the sllder-crank mechanism driving the flow was

increased thereby increasing Rema x at constant Rem. This increased the

crank radius to connecting rod length ratio (r/_). Figure 3-10 shows plots

of the displacement (x), the velocity (_) and the acceleration (_) of a

slider-crank mechanism. For r/_ - I/3 the acceleration, x, shows large

deviations from a sine function though the displacement, x, is nearly

slnusoidal. The turbulence structure is known to be sensitive to

acceleration and deceleration. It is, therefore, not possible to decide

whether deviations from sinusoidal, quasl-steady flow are due to changing

the drive geometry or due to the increase in the ratio of Rema x / Re_.

Ohmi et al. (1982) claimed the latter when they determined the line Rema X -

2800 /Re_ above which they predict turbulent flow to be quasi-steady.

They applied a quasi-steady I/7-power law velocity profile and observed that

it agreed well with their measured velocity profiled above this llne.

35
e

o

(7 90 ° 180 @ 2.F0 @ :360 @

m

rw

r/I =

rll_O

I I @

L_r/l=l/3

Figure 3-i0: Kinematics of the sllder-crank mechanism

36
Fluctuatin$ eddy viscosity model. Kita et al. (1980) found that the

Reynolds stresses In pulsating flow could not be adequately described using

a constant eddy viscosity. Therefore, they proposed a fluctuating eddy

viscosity model. Their five-layer model shows good agreement with

experimental results for pulsatlng flows; it was not tested for oscillating

flows.

Empirical pressure drop correlations. Taylor and Aghill (1984) took

data of pressure drop in oscillating flow through pipes of finite length £.

They plotted their data as a function of tlme-averaged Reynolds number for

various values of £/d. Their "friction factor" is based on a time-averaged

pressure drop and a tlme-averaged absolute value of the mean velocity. The

data indicate an increase of the friction factor by a factor of four over

the steady, unidirectional case.

From the previous discussion of pressure drop in laminar oscillating

pipe flow, it is clear that the acceleration and deceleration of the working

fluid influence the pressure drop in oscillating flow. Only if the results

were correlated in terms of a dimensionless frequency such as Re_, could

they be applied appropriately. Taylor and Aghili's (1984) paper does not

give sufficient data to calculate Re m.

3.6 Entrance and Exit Losses

Thus far, the discussion of pipe flow was restricted to fully developed

flow or, equivalently, infinitely long pipes. For finite pipes, entrance

effects may be important. The viscous losses are higher in the entrance

length of a pipe. Because of the spatial acceleration of the fluid,

_'V_ term in the momentum equation is not zero.

37
Goldberg (1958) showed that the hydrodynamic and the thermal entrance

length in hydraulically and thermally developing steady laminar flow is a

function of Reynolds number. Based on this observation, Charreyron (1984)

suggested that the entrance length varies over the cycle. Peacock and

Stalrmand (1983) hypothesized that the entrance length in laminar

oscillating flow will be shorter than in unidirectional, steady flow. Since

the velocity profiles tend to be flatter in oscillating flow, the velocity

profile of the oncoming flow may not have to change much in the entrance

region. Thelr hypothesis so far was not supported by experiments.

Disselhorst and van Wijngaarden (1980) studied separation near the

entrance of a tube under conditions of acoustic resonance. They found that

separation did not occur for high Strouhal numbers. During the inflow, a

boundary layer forms on the pipe walls; during the outflow, a Jet emerges

from the pipe. They observed and predicted that vortices that were shed

during the outflow would interfere with the formation of the boundary layer

during inflow.

3.7

to

Compressibility Effects

Compressibility effects in Stlrling engine heat exchangers may be due

(1) pressure recovery when the working fluid exits from a duct or a

heat exchanger tube into a large volume such as the cylinders,

(2) high Mach numbers in heat exchangers,

(3) travelling shock waves caused by the interaction of pressure waves

generated by the motion of the pistons,

(4) finite speed of pressure propagation.

38
Pressure recovery. The effect of the increase in pressure on the gas

density can be approximated by an adiabatic expansion. This is commonly

done by a compressibility correction factor ("expansion factor") for flow

through nozzles or sudden expansions (e.g., Perry etal. 1973, p. 5-11).

HiGh Mach numbers. If the Mach number of the fluid exiting from a heat

exchanger is one, a shock will form at the exit and the flow will be choked,

thereby limiting the mass flow rate. It will be shown later that these high

Maeh numbers are not expected for Stifling engines discussed in this report.

Travellin_ shocks. Research on non-linear acoustics in a resonance

tube shows that travelling shocks can form near resonance conditions at Math

numbers less than one (Mma x - 0.2) (Merkli and Thomann 1975 a and b). This

was predicted theoretically by Jeminez (1973). Resonance conditions,

however, are unlikely in Stirling engines because the flow length is

typically much less than half the acoustic wavelength (see Appendix E).

Travelling shocks can develop without resonance, however, by the following

mechanism.

When a piston moves towards top dead center, it causes a compression

wave to travel from the piston face. This wave travels through the

cylinder, duets and heat exchangers. The hlgh-pressure part of a wave

travels faster than the low-pressure part of the wave. Therefore, the

pressure wave steepens and may form a shock (Shapiro 1954, p. 949-950).

This process is affected by friction and heat transfer only through the

change in the local velocity of sound due to change in local fluid

temperature. A simple analysis of this type of shock formation is carried

out in Appendix E. This worst case estimate assumes that the working fluid

motion is driven by a piston with a velocity amplitude of Mma x - 0.15 (the

maximum value found in the data base, see Section 7). The fluid temperature

39
is assumed to vary linearly from Tc - 350K to Th - ]050K throughout the

engine and the frequency is the highest one found in the data base (SPDE-D),

f - ]05 Hz. Appendix E explains why this may be considered a worst case.

The results of this worst case show that formation of shocks in the

Stirling engines in thls data base Is not expected. The analysis is limited

in that it only includes right-travelllng waves and does not take into

account the interference of right- and left-travelling pressure waves from

the opposing pistons of an engine, which would distort the Math lines used

to predict shock incipience.

A more complete analysis of compressible flow in Stirling engines was

carried out by Organ (1982b) and modifications were proposed by Taylor and

Aghili (1984). Organ's analysis predicts that there is no incipient shock

for the special case of the air-charged engine discussed in his paper. This

section shows that Incipient shock formation is unlikely in the engines in

the data base.

Finite speed of pressure propagation. Pressure changes propagate with

finite speed, i.e., the sum of the velocity of the piston causing the

pressure change and the velocity of sound. Therefore, a phase lag is

expected between the pressures at both ends of a heat exchanger. If this

phase lag is a small part of the cycle, pressure propagation throughout the

heat exchangers may be considered instantaneous.

40
4. HEAT TRANSFER IN OSCILLATING PIPE FLOW

Since convective heat transfer depends on the velocity distribution,

the similarity parameters for fluid mechanics affect the heat transfer:

Remax, Hem, A H or £/d. In addition, similarity parameters from the energy

equation are important: the Prandtl number, Pr, and for high speed flows,

the Eckert number, Ec,the Mach number, M, and £h/d as a measure of the

thermal entrance length. The wall temperature, Tw, the bulk fluid

temperature at the hot end, Th, and at the cold end, T c, also control heat

transfer.

4]
4.1 Qualitative Considerations

Llmlted fluld dlsplacement. The 11mlted streamwlse displacement of

fluld in Stlrllng engines may llmlt heat transfer from the heat exchanger

wall or matrix as discussed In Section 2.1. Figure 4-I shows fluld

trajectories based on the plug flow assumption. While gh Is the

characteristic streamwlse length for the heat transfer problem, the flow

length & Is used In thls study because & , _h in most Stlrllng engines and

In many cases gh Is not documented.

_t

X

AKh<I

AR<I

AR-I

AR>I

Figure 4-I: Fluid displacement In heat exchangers.

42
Under the plug flow assumption, the AR values indicate:

AR < I Some fluid does not leave the heat

exchanger.

AR - I All fluid moves into and out of the

heat exchanger.

AR > I Some fluid moves through, some fluid

moves in and out of the heat exchanger.

Assessing plug flow, Organ (1975, p. 1016) argued that, in the absence

of axial mixing, the fluid that never leaves the heat exchanger in the ease

of AR < I could be considered additional dead volume. In turbulent flow,

the presence of axial mixing may invalidate Organ's model.

The velocity profiles for laminar flow in Figure 3-2 and In Appendix C

show that plug flow is not a good assumption unless Rem is very high. In

turbulent flow, plug flow may be a good assumption but axial mixing will

become important. The velocity profiles show that, in fully developed

osc111atlng pipe flow, the fluid displacement varies with radial distance.

Therefore, Organ's (1975) concept of moving temperature wavefronts is only a

rough approximation for turbulent flows. In fact, the radial variation in

fluid displacement leads to enhanced axial heat transfer as discussed in

Section 4.2.

Two temperature drivin_ potentials. Typical boundary conditions for a

tubular heat exchanger in a Stirllng engine are sketched in Figure 4-2.

43
Th To

Th > Tw > Tc

Figure 4-2: Boundary conditions for heat transfer in a heater.

During the hot blow, gas from the expansion space (at temperature Th) will

stream through the tube into the regenerator. The laminar velocity profiles

in Figure 3-2 show that, at higher values of Rem, the fluid near the tube

wall mayflow in one direction while the core moves into the opposite

direction (Rem - 30, crank angles 150° and 180°). For those parts of a

cycle where counterflow occurs, an unusual heat transfer situation exists:

the wall gives off heat to the fluid near the wall which enters the tube at

Tc from the regenerator. The fluid coming from the expansion space receives

heat from the near-wall-fluid in a counter-flow heat exchange process.

In this case, the heat transfer has two driving potentials, (T w - Tc)

and (T h - T c) where (Tw - T c) > (Th - Tc). It appears that backflow near

the wall may have a significant effect on heat transfer. Its influence

would depend on the degree of mixing within the near wall shear layer. The

concept of two driving potentials for heat transfer in oscillating flow is

44
also important because of the augmentation of heat transfer in the axial

direction discussed In the next section.

A situation of two temperature driving potentials also exists In film

cooling of gas turbine blades, where the temperature differences between the

hot gas in the free stream and the injected cooling air determines the heat

transfer in addition to the temperature difference between the free stream

and the blade surface. A concept llke the film cooling effectiveness may

prove useful for the interpretation of experimental results on heat transfer

in oscillating flow.

45
4.2 Axial Heat Transfer in Laminar Oscillating Flow

This section discusses how energy transport due to an axial temperature

gradient is enhanced by flow oscillation. Note that this augmentation does

not depend on turbulent cross-stream mixing and that the analysis only holds

for laminar pipe flow.

Physical interpretation. When fluid oscillates in a duct in the

presence of an axial temperature gradient, large oscillating temperature

gradients normal to the flow direction are generated. During half of the

cycle, cold fluid at the core of the duct extracts thermal energy from the

hot fluid in the boundary layer. During the other half of the cycle, hot

core fluid heats up cold fluid near the pipe wall. Thus heat is transferred

from the hot end to the cold end of the duct. For further physical

interpretation, see Kurzweg (1985b, p. 298).

Note that this physical interpretation, as well as the analyses

discussed below, hold for laminar flow. In turbulent flow, convective

eddy-transport in the cross-stream direction may reduce the oscillating

cross-stream temperature gradients that are necessary for the augmentation

of axial heat transfer.

Analysis. Watson (1983) analyzed diffusion in laminar, oscillating,

fully developed flow in impermeable ducts of arbitrary, uniform

cross-section. The analogous heat transfer situation is sketched in Figure

4-2 except that the walls are adiabatic. The pipe connects a high

temperature reservoir (TH) and a low temperature reservoir (TL). Assume

that end effects are negligible and that A R ( I. In a stagnant fluid,

assuming that there is no convection, the axial heat flow rate is due to

conduction:

46
_d 2 TH - TL

q - T kf g

Watson (1983) showed that the flux through a duct in which an incompressible

fluid oscillates in laminar flow can be expressed similarly as

- _d 2 TH - TL

--_-keff

where the apparent thermal conductivity

kef f - kf (I + Au)

depends on the augmentation coefficient Au, which is a function of Re_, Pr

and the dimensionless tidal volume

2_ Remax

VT " AR _ " _ Re

The augmentation coefficient is

Au - Au (Rem, Pr, VT) - VT 2 ¥ (Rem, Pr)

¥ is given in Appendix F for values of Pr and Re m that are relevant for

Stirling engine heat exchangers. Kurzweg (1985a, p. 461) provides a plot of

Au vs. JRem for Pr = 0.1, I., 10. He develops an asymptotic solution for

Rem Pr < _. Watson's (1983) exact solution was confirmed experimentally

by Joshl et al. (1983) for the case of a circular pipe. Gedeon (1986) and

Kurzweg (1985b) treat the case of flow between parallel plates. Kurzweg

(1985b) calculates the augmentation of axial heat transfer in the case where

the walls are diabatic, i.e. they undergo transient temperature fluctuations.

The walls contribute to heat transfer by absorbing thermal energy from the

hot fluid and giving off thermal energy to the cold fluid. Kurzweg's

(1985b) analysis shows that the augmentation of axial transport is a

function of Pr, Re_, VT, and the ratios of fluid to wall conductivity and

thermal diffusivity.

47
4.3 Experimental Data

Hwang and Dybbs (1980 and 1983) presented experimental heat transfer

results for oscillating flow in a tube. Figure 4-3 shows a schematic of

their experimental facility. Air was moved through a heater and cooler by

an oscillating piston into an adiabatic cold space. The piston moved

approximately slnusoldally. As it moved to the right, the air expanded in

the cooler and into the heater. The heat transfer was calculated by

measuring the temperature rise of the cooling water on the outside of the

test section. Thermocouples measured the (apparently time averaged) gas

temperatures at both ends of the cooler and the wall temperatures in the

cooler pipe.

216

(All dimensions in mm, net to scale.)

r "%_''_

-' 70

maximum

stroke

cold sp&ee cooler heater cylinder piston with

(test sectlon) slider crank

mechanism

Figure 4-3: Schematic of Hwang and Dybbs' (1980) experiment

A Nusselt number was then formed by the use of the log-mean-temperature

difference, the measured average heat flux, and the pipe diameter. The

results are replotted in Figure 4-4 by the present authors.

48
Nu

2O

15

10

®

®

®

&

i I ! i v I

,5 x 10 3

indicates the prediction

of transition according

to Ohmi et al. (1982)

2 standard deviations

of a data point

I

io _

! !

Re

max

' I

i. 5 x 10 _

Figure 4-4 : Hwang and Dybbs' (1980) data

49
Hwang and Dybbs (1980 and 1983) plotted Nu vs. Reos (Reos - Rema x /

(2w)) wlth Am - 2/A R as a parameter. In this experiment only two of the

parameters AR, Rema x and Rem could be specified independently. A R is

physically not significant since for most data points the gas will be blown

completely through the heat exchanger (A R > 2), thus Rema x and Re_ remain

as the only significant parameters.

The data in Figure 4-4 show certain trends:

(a) For most values of Rem, the data show a small slope for low

values of Rema x, then the slope increases with increasing

Re m.

(b) The data points with the higher slope tend to lle in one

common band.

(c) The individual data points typically scatter within a band

that is wider than two standard deviations of a single

data point; i.e., the scatter of the data cannot be accounted

for by the random error of single data points.

The lower Rema x data points of each curve may represent predominantly

laminar flow patterns over a cycle while the parts of the curves with

greater slope may represent turbulent flow during parts of the cycle. This

is argued by Hwang and Dybbs analogous to unidirectional steady flow where

Nu a Re I/3 in laminar flow and Nu m Re 0"8 in turbulent flow. In vlew of

statement (c) on the scatter of data no further conclusions are drawn.

The heat transfer In Hwang and Dybbs' experiment consists of two

different processes:

(a) blow of hot air through the cooler pipe Into the cold space.

During this blow the heat transfer may be approximated by

q"- h ATI m

where ATIm Is the log mean temperature difference

k Nu

and h - _ kg - conductivity of the gas

This can be computed if the instantaneous heat flux and

log-mean temperature difference (LMTD), or at least the time

average values for thls blow is known.

50
(b) blow of cool air through the cooler pipe into the heater.

Again, a value of Nubased on the tlme-average heat flux and

LMTDfor this part of the cycle could be calculated.

In Hwangand Dybbs' experiment, however, only the heat flux averaged

over the whole cycle and the LMTDbased on temperatures averaged over the

whole cycle are known. Thus the Nusselt numbercalculated is neither

representative of the hot blow nor of the cold blow, but is an average of

the two. The heat flux that Hwangand Dybbs used to calculate a Nusselt

numbermay be due to heat transfer during the hot blow only. The LMTDwhich

is apparently based on tlme-averaged temperatures probably underestimates

the temperature differences during hot blow. Thus it is difficult to draw

further conclusions from this data set.

Apparently, none of the transition predictions fit Hwang and Dybbs'

data (Figure 4-5). Possible explanations for this discrepancy are:

(I) The tube was very short (£/d - 28); therefore the transition

criteria which were presumably developed for fully developed

oscillating flow may not apply.

(2) The first data point on each curve may be taken from relative

amplitudes that are too small to measure the heat transfer

coefficient accurately (A R - 1.8).

Neither criterion may be sufficient to explain the change in

tlme-averaged heat transfer for the Hwang and Dybbs experiment because the

heat transfer is a function of:

(a) the location where turbulence occurs (e.g., near the wall or

in the core).

(b) the relative time during which laminar and turbulent flow are

present during the same cycle.

51
t _ _transition estimates based on

lOk_ ___ L:JHwang and Dybbs' (1980)data

Re _----transition observed by Ohmi et al. (1982)

max

___transition observed by Grassmann

/ and Tuma (1979)

10 3

• • • , , , • 12P

i0 I0 Re

Figure 4-5: Comparison of prediction and observation of transition.

lwabuchl and Kanzaka (1982) studied heat transfer in oscillating flow

In a test facility that was designed to obtain results for the design of a

specific prototype engine. A rough estimate indicates that they operated in

the laminar regime (11 < Rem < 83, 145 < Rema x < 733). They dld not

correlate their results in terms of a Reynolds number and a dimensionless

frequency but they studied the dependence of the heat transfer on parameters

such as rpm, mean pressure and phase difference between the two opposing

pistons. Their heat-transfer results are, therefore, not generally

applicable. They made the observation, that the choice of the phase

difference (90 ° or 180 °) did not change the heat transfer provided that the

52
Schmldt analysis was used to calculate the mass flow of gas through the heat

exchanger.

53
5. FLUID MECHANICS AND HEAT TRANSFER IN REGENERATORS

In the complex geometries of regenerators, flow separation is expected.

Therefore, analytical methods are not ap_llcable and experimental results

are required.

5.1 Steady Flow

Stacked, woven wire screens, randomly stacked metal fibres, folded

sheet metal, metal sponge and slntered metals are used as regenerator

matrices. Steady flow through these matrices has been studied extensively

in porous media research and in studies related to gas turbine and Stlrllng

engine regenerators.

Flow Regimes in Porous Media Flow

Dybbs et al. (1984) classified the steady flow regimes in porous media

according to a Reynolds number based on the average pore diameter.

Re < 1

1 < Re < 10

10 < Re < 175

175 < Re < 250

250 < Re < 300

Re > 300

Darcy flow regime

Boundary layers begin to develop on the pore wall

laminar flow

separated laminar flow, vortex shedding

separated flow with random wakes

turbulent flow

The Reynolds numbers separating these regimes are expected to hold for only

small values of Rem and only for materials similar to those used by Dybbs

et al. (1984), but they represent the only known work on transition that

54
would apply to regenerator matrices. The complex and irregular geometries

of porous media and the complex flow patterns in flows with Re > 175 make it

difficult to predict pressure drop and heat transfer with numerical methods.

Therefore experimental results are required for the prediction.

The Permeability Model

Beavers and Sparrow (1969) showed experimentally that the pressure drop

for steady flow of an incompressible fluid through a porous material can be

described by

-dp/dx - a_V + bpV 2

where :

ap b are constants to be determined;

V is the superficial velocity;

m

V m

pA

A is the frontal area of the porous material. In this equation, the

first term accounts for purely viscous pressure drop as it is described by

Darcy's law:

-dp/dx- _V/k

where the permeability, k, is a constant for each porous material•

Beavers and Sparrow were able to reduce pressure drop data for foamed nickel

(FOAMETAL by General Electric) of various geometries and porosities and for

wire screens by using the square-root of the permeability as the

characteristic length• The Reynolds number and friction factor were defined

as:

v_

Re -

55
Then the dimensionless pressure drop equation is:

f = l/Re + C

The values for C were found to be close to C = 0.074 for FOAMETAL and wire

screen, while a specimen of FELTMETAL (randomly stacked fibers, Huyck Mfg.

Co.) had a higher value of C = 0.132. There are free fiber ends in

FELTMETAL which are not found in the other materials. This structural

difference may be the reason for the difference in C.

To see whether the results obtained by Beavers and Sparrow are relevant

for Stifling engine flow conditions, the present authors calculated the

Reynolds number for the wire screen specimen, which Beavers and Sparrow used.

This Reynolds number, comparable to Remax, ranged from about 64 to I04.

In order to use the friction factor correlation

f - I/Re + 0.074 ,

the permeability must be known. This quantity is obtained experimentally.

The pressure drop of creeping flow is measured and the permeability is

calculated from the experimental data using Darcy's law as given above.

Dybbs et al. (1984) studied steady flow in porous media. They observed

that at higher velocities than those characteristic of Darcy flow, the

friction factor is a function of a Reynolds number based on the hydraulic

diameter and on the ratio of length to diameter of a typical pore. Their

explanation was that the flow must develop anew in every pore. For the

prediction of pressure drop and heat transfer, this may mean that the

length-to-diameter ratio of a typical pore should be included as an

additional geometric parameter. The permeability cannot represent the

effect of the pore shapes properly because it is measured in the

56
non-inertial Darcy regime. This is one reason why only geometrically

similar materials have the same C-value as observed by Beavers and Sparrow

(1969).

Macdonald et al. (1979) reviewed research on flow in porous media

consisting of spherical beads, cylindrical fibers, sand, gravel, and many

others. They obtained an equation that predicts pressure drop for a wide

range of materials within ±50%.

Joseph et al. (1982) showed that the pressure drop in a bed of packed

spheres can be related to the drag on a single sphere. They also lent

theoretical support to the velocity-squared term proposed by Beavers and

Sparrow (1969).

Beavers et al. (1973) studied the influence of the shroud bounding

porous media. They found that the pressure drop in beds of packed spheres

was influenced by the walls for shroud-diameters as small as 40 sphere

diameters.

Friction Factor Correlations

Kays and London (1984, p. 149) provide widely used correlations for

flow through stacked screens. Walker and Vasishta (1971) present

experimental data for dense-mesh wire screens. Miyabe et al. (1982) provide

additional experimental data for flow through stacked screens, Takahashi et

al. (1984) present data for foamed-metal matrices. Chen and Griffin (1983)

derived an empirical equation based on the Kays and London (1964) data that

more accurately fits the data. k comprehensive review of regenerator

pressure drop and heat transfer correlations for steady flow was prepared by

Finegold and Sterrett (1978). The disadvantage of these correlations is

that they do not collapse data as well as the correlation proposed by

57
Beavers and Sparrow (1969) because porosity remains as a parameter in

addition to the Reynolds number. The advantage is that they were obtained

for gas flows so that somecompressibility effects such as reduced effective

flow area, which is discussed below, are included.

Compressibility effects in flow through screens. Organ (1984)

recommended that the Mach number be included as a correlating parameter for

the prediction of pressure drop in regenerators. His suggestion is based on

earlier work by Benson and Baruah (1965) and Pinker and Herbert (1967) where

the flow of air through a single screen was investigated.

Benson and Baruah (1965) stated in their conclusions (p. 458) that "The

resistance coefficient of a gauze is a function of Mach number, Reynolds

number and solidity," as quoted by Organ, solidity - l-porosity. In the

same paper they plot the resistance coefficient of screens (with porosities

of 0.47...0.67) versus downstream Mach numbers. For low Mach numbers (M <

0.1) Benson and Baruah's (1965) results show that the resistance coefficient

is virtually independent of Mach number. Benson and Baruah (1965) also

state in their conclusions (p. 458) that a "gauze behaves like a crude

nozzle. There is a considerable entropy change in a flow across it."

According to Pinker and Herbert (1967, p. 16 and p. ?) this "nozzle"

develops at Mach numbers of the oncoming flow of M - 0.2 in the duct and of

M - 0.38 for a porosity of 0.53. The duct Mach number required for choking

increases with porosity.

Effective Flow Area in Compressible Flow

To calculate the mass flow rate through a nozzle, the flow area is

multiplied by an expansion factor (see e.g., Perry et al. 1973, P. 5-11).

It accounts for the reduction in effective flow area due to the adiabatic

58
expansion from the upstream to the downstreampressure. It is expected that

a slmllar effect can be observed when flow passes through a single screen.

It is not likely that this effect will be significant for flow through

stacked screens because the pressure differences across individual screens

will be small and heat transfer from the matrix to the gas will be possible.

Choking and Pressure Drop In Compressible Flow

Beavers and Sparrow (1971) calculated the condltlons under which the

flow into a porous material would be choked based on Isentroplc flow.

According to Benson and Baruah (1965, p. 458), a considerable change in

entropy takes place when a gas passes through at higher Math numbers. Thls

entropy change may not be significant for the low Mach numbers encountered

in Stlrllng engines, however. _

Beavers and Sparrow (1971, p. 1856) also analyzed gas flow within a

porous matrix using /k'(the square-root of the permeability) as the length

scale. They calculated the maximum length £max of porous material before

the exlt flow would be choked. A regenerator operated at hlgh Mach number,

low Reynolds number and wlth a high _/k-ls most likely to choke. Even in

the extreme case of the cold blow through the 4L23 regenerator £max//_ >

I0£/_. Therefore choking is not expected in regenerators.

This analysis is tentative because:

(I) The permeabllltles were calculated from the comparison of

permeability and hydraulic diameter for one specimen that Beavers and

Sparrow (1969) used; to calculate the permeabillties for other wire-mesh

regenerators a linear relationship between hydraulic diameter and the

square-root of the permeability was assumed.

59
(2) the graphs used to calculate _max/J_-in Beavers and Sparrow (1971)

were plotted...

(a) for air only.

(b) for discrete values of Reynolds number.

(c) for discrete values of porosity.

Heat transfer correlations. Kays and London (1984, p. 149), Walker and

Vasishta (1971), Finegold and Sterrett (1978), and Miyabe et al. (1982)

provide heat transfer correlations for packed woven wire screens. These

correlations are derived from experimental data that were obtained by the

single blow technique described in Kays and London (1984, pp. 154-155).

Takahachi et al. (1984) calculated heat-transfer correlations from

oscillating flow data. The use of _k-as the characteristic length instead

of the hydraulic diameter may eliminate the porosity as the independent

parameter in heat transfer correlations. This remains to be shown

experimentally.

5.2 Unsteady Flow and Heat Transfer in Regenerators

Galitseiskii and Ushakov {1981) studied heat transfer augmentation due

to reversing pulsatile flow in porous media. They observed a resonant

phenomenon when the diameters of the stationary vortices (generated by the

flow over the elements of the porous structure - e.g. the wires of the

screen) are comparable to the dimensions of the secondary vortices generated

by the flow oscillations. The experiments were performed for 0.6 < Rema x <

6 and 6.5 < Re m < I00, a domain that does not seem applicable to

regenerator flow conditions.

Kim (1970) studied oscillating flow through a regenerator consisting of

packed spheres. The reduction of his experimental data was based on the

60
assumption that the momentum equation could be considered quasi-steady i.e.,

the pressure drop is in phase with the mass flow rate. He showed by

order-of-magnitude estimates that this was Justified (p. 130). He

calculated friction factors for gas flow through a bed of packed spheres

that were approximately 20% higher than the corresponding steady values.

The range of frequencies covered less than a decade. Since the stroke of

his apparatus was fixed, the dimensionless frequency of his experiments was

proportional to the peak Reynolds number. Therefore, his data do not allow

conclusions about the frequency dependence of the friction factor. His

results do, however, suggest that fluid friction increases due to flow

oscillation. Because the friction factor is higher in oscillating flows

than in steady flow, it must depend on frequency (Rem). Whether the same

is true for wire-screen regenerators must be answered by experiment.

5.3 Regenerator Theory

Stirllng engine simulation typically uses a discretization of the

engine volume. If the regenerator is subdivided in several control volumes

it is not necessary to use regenerator effectiveness to predict heat

transfer to and from the regenerator (Walker 1980, p. 140-149). Given an

adequate regenerator theory, however, measured effectiveness values can be

used to determine volume-averaged heat transfer coefficients of regenerators.

This method has been used successfully for gas-turbine regenerators (Kays

and London 1984, p. 154-155).

Rice et al. (1985) proposed a theory that takes into account the

relatively short blow periods in Stifling engines. Willmot and Hinchcliffe

(1976) and Harness and Neumann (1979) proposed methods to take "holdup" into

account, i.e., the fact that some fluid may not leave the regenerator by

61
convection (AR < I). These approaches seemmore promising than the

application of standard regenerator theories (e.g. Shah 1981, p. 721-763 or

Kays and London 1984, p. 79-89) to Stirling engine conditions, e.g. by Rice

et al. (1983) and Miyabe et al. (1982).

62
6. STIRLINGENGINEDATABASE

Representative values of the Similarity parameters describing the

operating conditions of the heat exchangers of 11Stirllng engines are

obtained to form a Stlrling engine heat exchanger data base.

6.1 Evaluatlon of Slmllarlty Parameters

Schmldt analysls. The Instantaneous mean mass veloclty pu m is

calculated using the Schmldt Isothermal analysis (Urlell and Berchowitz

1984, Appendlx AI, p. 152)• The following assumptions are made for the

analysis:

I • The engine consists of three isothermal volumes:

The hot space (at Th) consists of expansion space, connecting

duct, heater, I/2 heater-regenerator duct. The regenerator

volume (at T r) consists of I/2 heater-regenerator duct, the

regenerator and 7/2 regenerator-cooler duct. The cold space

(at T c) consists of compression space, connecting duct,

cooler, I/2 cooler-regenerator duct. Tr is the mean effective

regenerator temperature (Urlell and Berchowltz 1984, Appendix

A4, p. 158)

Th - T e

Tr - in(rh/Tc )

2. The pressure is uniform throughout the engine.

3. The ideal gas law holds.

4. The variations of the compression and the expansion volume are

slnusoldal.

5. There Is no working gas leakage.

6. The engine operates at constant speed in a cyclic steady

state.

7. The kinetic and potential energies of the gas are negligible.

Isothermal analysis. The isothermal analysis is based on the same

assumptions as the Schmldt analysis except point 4. The variations of

compression and expansion volumes are calculated from the kinematics of the

drive mechanism. Appendix A shows Reynolds number variations calculated for

63
a rhombic drive engine (GM GPU-3) with Schmidt (Figure A-S) and isothermal

analysis (Figure A-I_) (Youssef 1986). The two analyses differ in the shape

of the Reynolds number transients. The difference in amplitude of

approximately 20 percent is acceptable because this data base is only used

to establish approximate ranges of heat exchanger operating conditions.

Reynolds and Mach number. The thermodynamic analysis is used to

calculate the changes of mass in various control volumes of the engine.

From the mass changes in the control volumes, mass flow rates are

calculated, and from these Reynolds and Mach numbers, um, the mean

velocity, is the velocity averaged over the flow area, e.g. over the

cross-sectional area of a duct. Figure A-I shows the variation of the

Reynolds numbers at the inlet and outlet of a GPU-3 heater calculated by the

isothermal analysis. The values at inlet and outlet differ because working

gas is stored in the heat exchanger during the compression and released

during the expansion part of the cycle. This changes the mass flux term 0u m

in the Reynolds number. Therefore, the maximum Reynolds number at each

operating point is represented by a range of values. This range represents

the variation throughout the heat exchanger due to asymmetric volume

variations in storage and release of working gas during compression and

expansion (see Appendix A). The dynamic viscosity is evaluated at the

temperature of the corresponding control volume, the density is calculated

from the ideal gas law, and the speed of sound is evaluated assuming

constant specific heats, a = Y_. The hydraulic diameter used to calculate

Rema x for non-tubular geometries (rectangular ducts) is defined as the ratio

of volume to heat transfer area of the heat exchanger.

Dimensionless frequency Re_ and relative amplitude AR. The

dimensionless frequency is evaluated at the mean pressure and the

64
temperature of the respective heat exchanger. For regenerators, the

hydraulic diameter is used as a length scale. The relative amplitude of

fluid motion, AR, is evaluated by integrating the mass flow at the heat

exchanger inlet or outlet between two flow reversals. For the evaluation of

the displacement of a fluid element between flow reversals the fluid is

assumedto moveinto and out of the heat exchangers as a plug.

6.2 Documented Engines

The choice of documented flow conditions was restricted by the data

available. Table 6-I lists (alphabetically by abbreviation) the name of the

engines included in the data base, a brief description of the operating

conditions and the reference used. The amplitudes of the pistons and

displacers of the free piston engines were not obtained from the analysis

but experimental values were taken from the references.

The SPDE "design" point (SPDE-D) was calculated from phase angle and

amplitudes, measured in the trial run represented by "SPDE-T" but using

design pressure and frequency. A more realistic approximation to the

intended operating conditions was calculated from data provided by Tew

(1986) and is labeled "SPDE-O."

65
Table 6-I: Documented Engines

Engine Type Operating Conditions:

Speed or frequency

Mean pressure

Working fluid

Reference Abbreviation

Kinematic Engines

General Motors

GPU3

Mechanical

Technologies, Inc.

MODI (upgraded)

United Stirling AB

P40

Technical

University of

Denmark,

Stirling Total

Energy System

UK Consortium

Stirling Engine

(a-configuration)

Ford Phillips

4-215

General Motors

4L23

2500 rpm

41 bar

He

4000 rpm

150 bar

H2

4000 rpm

150 bar

H2

1500 rpm

100 bar

He

3000 rpm

150 bar

He

4000 rpm

200 bar

H2

2000 rpm

103 bar

H2

Urieli and Berchowitz

(1984), pp. 37 and 39

Richey (1986)

Tew (1983)

pp. 80-86

Andersen (1979)

pp. 50-52

Dunn et al. (1982)

p. 68

Urieli and Berchowitz

(1984), p. 30

Martini (1982)

p. 32

GPU3

MODI

P40

STES

UK

4-215

4L23

66
Table 6-I: Documented Engines (continued)

Engine Type Operating Conditions:

Speed or frequency

Mean pressure

Working fluid

Mechanical 58 Hz

Technologies, Inc. 60 bar

Engineering He

Model

Sunpower, Inc. 60 Hz

3 kW Generator 25 bar

Set air

Sunpower, Inc. 30 Hz

RE-IO00 70 bar

He

Mechanical

Technologies, Inc.

Space Power

Demonstrator

Engine

Reference

Free Piston Engines

Dochat (1985)

trial parameters:

73 Hz

75 bar

He

extrapolated to design

conditions:

105 Hz

150 bar

He

intended operation:

105Hz

150 bar

He

Berchowitz (1985)

Schreiber (1983)

Dochat (1985)

Doehat (1985)

Tew (1986)

Abbreviation

EM

Genset

RE1000

SPDE-T

SPDE-D

SPDE-O

67
7. RESULTS AND DISCUSSION

The operating conditions of the heat exchangers in the Stlrllng engine

data base are presented in terms of similarity parameters and conclusions

are drawn about the conditions of fluid mechanics and heat transfer in

heaters and coolers and in regenerators. The significance of various

similarity parameters is discussed. Plots of heat exchanger operating

conditions in terms of Remax, Re m and AR are used.

7.1 Heaters and Coolers

Entrance and exit losses. Table 7-I gives a list of rough estimates of

the points of the total heat exchanger pressure drop occurring at the heat

exchanger entrance, in the duct of the heat exchanger, and at the exit. The

estimates are based on the assumption of steady, unidirectional, turbulent

flow in a rough pipe. Between 15% and 50% of the heat exchanger pressure

drop occur at the pipe entrance and exit. Therefore entrance and exit

effects are important. They could be estimated from an experiment such as

the one by Taylor and Aghili (1984) and should be correlated in terms of the

area ratios of contraction and expansion and Rema x and Re m.

68
Engine

Abbreviation

H - heater

C - cooler

GPU3 H

C

MODI H

C

P40 H

C

STES H

C

UK H

C

4-215 H

C

4L23 H

C

EM H

C

Genset H

C

REIO00 H

C

SPDE H

C

Portion of the Pressure Drop

(in percent) occurring...

at the I in the

entrance I pipe

68

53

71

71

70

67

56

5O

85

47

75

71

7O

74

67

67

62

62

68

72

65

62

15

22

14

14

15

16

22

24

7

24

12

14

14

13

16

16

18

16

15

13

17

18

at the

exlt

17

25

15

15

15

17

22

26

8

29

13

15

16

13

17

17

2O

22

17

15

18

2O

Table 7-I: Estimated pressure drop distribution

in heaters and coolers.

Estimated area ratios for the heater-regenerator and cooler-regenerator

flow are listed in Table 7-2 to provide the similarity parameters for

experiments on entrance and exit losses in oscillating flow.

69
Engine

Abbreviation

GPU3

MODI

P40

STES

UK

4-125

4L23

EM

Genset

REIO00

SPDE

Heater-Regenerator

11.

21.

40.

32.

29.

30.

9.2

13.

7.6

7.8

13.

Cooler-regenerator

11.

13.

17.

16.

5.9

18.

18.

9.4

3.0

4.5

9.0

Table 7-2: Ratios of regenerator housing area to

heater/cooler flow area.

Developing profiles. Table 7-3 lists _/d values. For heaters and

coolers with rectangular cross-section, the hydraulic diameter is used. In

steady turbulent flow, the hydrodynamic entrance length is independent of

Reynolds number and between 10 and 60 pipe diameters long (Incropera and

DeWitt 1981, p. 377). Therefore, developing velocity profiles may be

expected throughout many heaters and coolers.

70
Engine

Abbreviation

GPU3

MODI

P40

STES

UK

4-215

4L23

EM

Genset

REIO00

SPDE

Heater

81 .I

96.9

93.7

51.0

219.0

115.0

88.6

81.4

59.1

88.5

71 .O

C_/d)

Cooler

42.7

92.0

80.O

38.9

30.8

96.7

112.0

75.9

50.9

77.6

62.5

Table 7-3: Length to diameter ratios of heaters

and coolers.

Flow patterns and axial<heat transfer. Figure 7-I shows that the

operating conditions of heaters and coolers fall within two decades of

dimensionless frequency. The amplitude Reynolds numbers are all above the

Reynolds number at which transition occurs in steady unidirectional flow.

Most heaters and coolers operate in the turbulent flow regime but many are

near transition. Therefore, it is important to understand the transition

process and the characteristics of fluid mechanics and heat transfer in

turbulent oscillating flow. Three heat exchangers operated under

transitional conditions: the REIO00 cooler and the SPDE heater and cooler.

The SPDE cooler is shifted from turbulent to transitional operation as the

engine frequency is reduced. The same is expected if the Genset cooler or

heater is operated at frequencies below the design point. Based on the

discussion above, higher power dissipation and pressure drop than predicted

by quasi-steady analysis are expected in the laminar and transitional

regimes. Due to fluid inertia, there will be a phase shift between pressure

7]
and mean velocity. Therefore, the instantaneous pressure drop cannot be

determined from the instantaneous velocity and acceleration alone.

The trial and design operating conditions of the SPDE fall on a

straight line of slope I in the double logarithmic coordinates of Figures

7-I and 7-3. The ratio of the two similarity parameters is

Re _d _ _d

Since d is constant and Uma x is proportional to m for any one engine, the

ratio is constant. Therefore Rema x is proportional to Re m if operating

conditions are extrapolated assuming that only m changes.

72
to_

turbulent

Remax

10 2 103

I I0 Rew

Figure 7-1: Re _. Re £or heaters and coolers

73
Figure 7-2 shows that the four heat exchangers in the transitional

regime are operating with AR < I; therefore, one might think that axial

augmentation would be important. Estimates of this heat transfer

augmentation for the SPDE heater and cooler indicate that the axial heat

transfer is only about 0.1 - 1.0% of the heat transferred by the heat

exchanger, however. This estimate was made using Watson's (1983) analysis

and is based on the assumtlon that the pipe walls are adiabatic. Further

work Is required to extend this analysis to non-adiabatic wall conditions as

in the Stlrllng engine.

Figures 7-2 and 7-4 show that the relative amplitude of fluid

displacement AR is the same for the trial and the design conditions, while

the dimensionless frequency Re_ is different. The trial and design

conditions differ in the operating frequency and in the fluid density

because of the different mean pressures. Therefore

Re =

4v

is affected while AR is not affected.

74
IO

AR

I.O

O.I

IO

GPU5 i_ISTES

I T °_

_ REIOOO SPDE-O

SPDE-T_

El COOLER

O HEATER

IO 2 ,o_

Figure 7-2: A_ vs. Re for heaters and coolers

K

75
Compressibility effects. Table 7-4 shows that the Mach numbers are

small in heaters and coolers, Mma x < 0.075, with the exception of the UK

heater (Mma x - 0.15). The density variation due to high velocity in the

latter case is (Schllchtlng 1979, p. 10):

P = _ Mmax2 = 0.012

Therefore, hlgh density variations due to hlgh fluid velocity are not

expected in the coolers and heaters of these Stlrllng engines.

Engine

Abbreviation

GPU3

MODI

P40

STES

UK

4-215

4L23

EM

Genset

REI000

SPDE-T

SPDE-D

Highest Mach Number in the

Heater

0.042

0.067

0.064

0.035

0.153

0.072

0.027

0.020

0.038

0.007

0.008

0.011

Cooler

0.034

0.046

0.030

0.015

0.031

0.043

0.040

0.029

O.026 '

0.020

0.008

0.009

Table 7-4: Mach numbers in heaters and coolers.

76
7.2 Regenerators

Figure 7-3 shows that the flow pattern in most regenerators may be

expected to be complex. According to the observations by Dybbs et al.

(1984), the flow would be turbulent or transitional for 8 of the 12

regenerator operating points. Since their observations were made in steadystate experiments, they may not hold in oscillating flow or may hold only

for low Re m and will probably only apply to matrices similar to those

studied by Dybbs et ai.(1984).

Six of the 11 engines shown In Figure 7-4 have A R < I which may limit

the heat transfer from the matrix to the adjacent heater and cooler

("holdup").

Math numbers in the regenerators are Mma x < 0.0] (Table 7-5). Choking

of the regenerators is therefore not expected. At these low Math numbers,

the pressure drop in the regenerator matrices is expected to be independent

of Math number, according to the results of Beavers and Sparrow (1971) and

Benson and Baruah (]965) discussed in section 5.]. Since porosities of

regenerators typically exceed 0.5 and Mach numbers are less than 0.2,

choking in regenerator screens is not expected according to experimental

results by Pinker and Herbert (1967, p. 16 and 17) which were discussed in

Section 5.1.

?7
104

, , , I,,,, I

steady flow

regimes according

to Dybbs et 01.(1984)

! | • = I .

,o3 m

R

m

B

n

Re _

max

2

I0 --

mm_

m

turbulent

m

transitional

laminar

GPU3

•. MODT

5-r 4L23 .. 4-21 RE I000 --

• SPD

m •

STES

/

/

f

/

/

/

• J

/ /

/ /

/ /

J /

.II /

/

.=

SPDE-T

EM

SPDE-D

• m

Genset

7'1

IUK

I

m

m

m

i

u

I0

.01

I I I I ! ! I I't

I0

Flgure 7-3: Re

max

vs. Re for regenerators

78
I0

AR

1.0

0.1

0.01

GPU3 ISTE s I 4L23

i,,.2, 5M°°II ,E,ooo lIp,o

S P DE-IoI IEM

......... SPDE-D Genset

0.1 1.0 I 0

Figure 7-4: AR vs. Re for regenerators

79
Engine

Abbreviation

Highest Math Number

in the Regenerator

GPU3 0.0078

MODI O. O093

P40 0.0061

STES 0.0031

UK O.0103

4-215 O. 0055

4L23 0.0047

EM O. 0056

Gen se t 0. O096

RE 1000 O. 0038

SPDE-T 0.0017

SPDE-D 0.0024

Table 7-5: Mach numbers in regenerators.

7.3 Similarity Parameters

After a review o£ the operating conditions o£ Stlrllng engine heat

exchangers, several similarity parameters can be eliminated. Eckert numbers

were calculated from

T

Sc - (_ - I) -r_M_ax 2

A*--

They range from 0.0066 (SPDE-D, heater) over 0.025 (MODI, heater) to 0.13

(UK heater). Therefore, viscous dissipation is not considered to contribute

significantly to the heat transfer, with the exception of the UK heater.

The exact numerical 7alue of the Eckert number is not important, provided

that it is less than - O.i.

A similar argument can be made for the Mach number, because Tables 7-4

and 7-5 show that they are small, with the possible exception of the UK

heater. Thus similarity wlth respect to Mach number is not required as long

as Mma x < 0.1. Organ (1982a) challenged this argument with experimental

80
pressure drop results, but his data seem to support a strong frequency -

(rpm-) dependence of pumping power.

Pressure propagates through the heat exchangers fast compared to the

cycle time. Table 7-6 provides estimates of the time required for

propagation of pressure through the heat exchangers. It is assumed that

pressure waves travel with the speed of sound which underestimates the

propagation speed slightly. The data are given in degrees of crank angle.

In all heat exchangers but the UK and 4-215 heater, pressure propagates

through the individual heat exchangers within 1/100 of the cycle time.

Therefore, effects due to the finite propagation speed of pressure are not

expected to be significant.

Engine Heater Cooler ReEen.

GPU-3 2.0 0.69 0.24

MODI 3.3 1.5 0.83

P40 2.9 1.3 0.51

STES 0.76 0.58 0.094

UK 5.2 1.6 0.64

4-215 4.5 I .5 0.43

4L23 2.0 1.1 0.16

EM I. 2 2.0 0.86

Genset 2.8 6.2 3.1

REIO00 1.2 0.82 0.52

SPDE-T 1.7 2.4 0.54

SPDE-D 2.4 3.5 0.78

Table 7-6: Time estimates for pressure wave propagation

through heat exchangers in degrees of crank angle.

8]
. CONCLUSIONS AND RECOMMENDATIONS

In this report, important similarity parameters are introduced:

Rema x like Reynolds number in steady flow

Re m measure of unsteadiness

AR, ARh relative amplitude, important for heat transfer

_ld, _hld developing length effects

Length ratios maintain geometric similarity.

De like Rema x for curved pipes

Area ratios maintain similarity for entrance and exit losses.

For porous media, the square-root of the permeability, _, may be

the appropriate length scale to be used in 8ema x and Re m.

Previous research results are discussed and 11 Stirling engine operating

points are plotted in terms of the parameters.

Due to the acceleration and deceleration of the fluid, velocity

profiles and stability of laminar oscillating flow differ from steady

laminar flow. Transition occurs at higher Rema x as Re_ is increased.

Adequate turbulence models or correlations for turbulent oscillating flow do

not exist. Results of past research on heat transfer are not generally

applicable except the results for enhanced axial transport in oscillating

laminar flow. It was shown that the heat exchangers of several engines are

operating in or near the laminar-to-turbulent transition region while most

heat exchangers operate in the turbulent regime. Several heat exchangers

are long relative to the fluid displacement, A R < I. This means that some

heat exchangers may add to the engine dead volume.

The effect of compressibility of the working gas on pressure recovery

can be accounted for by assuming an adiabatic expansion at the heat

82
exchanger and duct outlets. Choking is not expected due to the low Mach

numbers encountered. Incipient shock formation was shown to be unlikely

using a worst-case analysis. Pressure waves were calculated to pass through

the heat exchangers quickly compared to the cycle time.

A review of the literature has shown that immediate attention is needed

in the following areas to assess the trade-offs between heat transfer and

viscous dissipation in oscillating flow:

I. The process of transition must be understood more clearly. Work on

this topic in tubular geometries will improve the understanding of

heat transfer and pressure drop mechanisms in the heaters and

coolers of Stirling engines.

2. The means by which turbulence structure Is affected by flow

oscillation must be better understood, possibly leading to ways of

modifying standard turbulence models so that turbulent oscillating

flows can be accurately predicted.

3. The effects of thermal and hydrodynamic entrance lengths on heat

transfer and pressure drop in tubes and regenerator matrices must

be better understood.

4. A test of the permeability model should be made. It may provide a

more general characterization of regenerator matrices than the

characterization by hydraulic radius and porosity currently used.

Once these topics have been addressed, non-sinusoidal oscillating flow

should be studied experimentally to determine how general the results of I-4

are. A computational program can then use the experimental data from I-3 to

calculate heat transfer in oscillating flow with compression and expansion

of the working gas. Its results should be confirmed experimentally.

83
ACKNOWLEDGEMENTS

This work was supported by NASA-Lewis Research Center under NASA Grant

NAG 3-598. The authors thank grant monitor James E. Dudenhoefer for his

guidance. Comments and suggestions by Roy Tew (LeRC), Manmohan Dhar and

Anthony Bright (MTI), and David Gedeon were valuable in revising this report.

We thank Youssef Sadek and Keith Shodeen for their help with some

calculations, and N.E. Andersen, D. Berchowitz, G. Dochat and A. Richey for

engine operating data.

84
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of Flow and Viscous Drag in Arteries when the Pressure Gradient is

Known." Journal for Physiology, Vol. 127, pp. 553-563.

Yamane, R., Oshima, S., Sudo, K., Sumida, M., Okamoto, N. and Kizaki,

M. (1985): "Study of Oscillatory Flow in a Curved Channel." Nihon

Kikaigakkal (Bulletin of the JSME), Vol. 28, No. 237, March.

¥oussef, Sadek (1986): "Isothermal Analysis of a Rhomblc Drive

Stlrllng Engine and Comparison with Schmidt Analysis." Internal

Report, Mechanical Engineering Department, University of Minnesota,

Minneapolis.

Zielke, W' (1968): "Frequency-Dependent Friction in Transient Pipe

Flow." Journal of Basic Engineering, Trans. ASME, Series D, Vol. 90,

No. I, pp. IO9-115, March.

92

(k _
APPENDIX A: A SURVEY OF PRESSURE, REYNOLDS-NUMBER AND MACH NUMBER

VARIAT IONS

Figures of the pressure, Reynolds number and some Mach number

variations are provided to give the reader the opportunity to assess the

quality of the data base discussed in Chapter 7. Figures AI and A2 show

results of the isothermal analysis of a GPU-3 (Youssef 1988). Figures A3

through A14 were obtained by Scl_mldt analysis.

- Flow is positive towards the cold end.

- full line - hot end

- dashed line - cold end

93
_s

I,, =.,, ,,'° N', --!o ,_'o , =!° m!° m', -." ,.'

CI_NK RNTJLE CI_NK RN_LE

Figure A-l: Results of the isothermal analysls of a GPU 3:

pressure and Reynolds numbers

94
Qj

i'

?

,x ,y i'

', _ _',"

I

I:RRNK RNE,LE

i

/ \

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,,!, ,,,!, ,,,,?, --!,

_K RNSLE

I

I X

.,o/ ,'/'%

_'l _/ \

+/_ ,'/

CRRNK RNE,LE

Figure A-2 : Results of the isothermal analysis :of a GPU 3:

Mach numbers

95
Figures A-3 through A-14:

Schmidt Analysis Results

I

m.o

/ •

!., o!0 _!, N!0 _?0 --', I.,,

CflFINK PtN_LE

%1

0!, m!, N!, --"

Ceq_lKP,N61.E

q.,.

Figure A-3: GPU 3

96
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98

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100
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101
i !.

' i !

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102
OK POOk _:;/,,U[TY

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Nil moll

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103
Figure A-II: Genset

104

ORIGIT_$AL PAGE IS

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DRIGINAL v* ,_

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105
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106

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107
APPENDIX B: DERIVATION OF SIMILARITY PARAMETERS

Normalization of the momentum equations

To normalize the momentum equation, the following dimensionless

variables are used:

4_

_ X

X m --

L

,.F

÷* U

U w --

U

*=2_ P

Po

*=.E_ P

rb

m

v* - -_

P

Neglecting gravity, the momentum equation Is:

Bt p p

to_Y"÷ _-_*.v*_*-

U2 V*_r.*" + _ u _ * v*'_*

L p po _1" p

to Po _ + _o _o P

108
For the oscillating pipe flow problem, the following scales are used:

length scale

velocity scale

time scale

L = d/2

U - Um,ma x

to - 11w

pipe radius

amplitude of the mean velocity

period of oscillation / 2_

4 vo at 2 vo 2 vo p

Re a_" R-Smax _'. V'_" _ V__ + v'V" '_"

= _--_' + 2 " - 2 p

Um:m_ x d and Re . u d 2

where Remax " v _ 4--'_"

The dimensionless frequency Re_ appears as a coefficient of the unsteady

inertia term in the normalized Navler-Stokes equation and the amplitude

Reynolds number Rema X appears as a coefficient of the steady inertia term.

Dividing by the maximum Reynolds number, the following form is obtained:

,.,d a;" V__ 2_o _'V'';'

2 Um,max at" + _"V*_" - - + P Um,max d

Str a_/" _*" V*_* V*om 2

2 at"÷ " - _ ÷ Remax

In this form, the Strouhal number and the Reynolds number appear as the

coefficients:

Um.m_w d W d

Rema x - and Str -

Vo Um,max

lOg
Normalization of the energy equation

The energy equation is:

BT Bp ÷

pop (-_ + _-9T) _ + 9p + V'(kVI') + #

where $ is the dissipation function, using Stokes' hypothesis (e.g.

Schlichting 1979, p.60):

_u

# - TiJ @x i " g(_x-_) =

The normalized variables are chosen as:

t N - =t

d/2

,¢,

"¢',R U

U -

Um, max

* p - Pm_nn

P " =

PO um,max

T* = T - Tp,

Th - T O

V* = d/2 V

P " P/Po

ki . k/k o

= _/_o

Cp* = Cp/Cpo

i au m 2

With these variables, the energy equation is:

110
p, (m(T h TO ) 8T _ V* Pc °po Cp* - _ + (Um, max _*" d/---2T*) (T h - Te)) -

Pc Um,max 2 + (Um,max _*" V* pS) Pc Um 2

d/---2 ,max

+ (T h - TO ) ko _-7_ V* • (k* --V*d/2 T*) ÷ Pc (ud__2 ), #*

, [ m d/2 aT* _*.V*T* d/2 a *

P °P' us,ma-q_'* ] . epo(T um._h - TO ) _t ÷ CpoUm'm_'2 (Th - T c) _'V*p"

+ kn I V*'(k*V*T*) + l_n_ (Th To) PoCp d/2 Um,ma x Poepo -

p*Cp* r2t----_-V Re 3T* ÷ _*"V'T* ]- _2 Re Ec _t* ÷ Eo_*-v*p* Remax Remax

+ I V* "<h*V*T*) + Ec @,

Pr Rema x Rema x

Finally, the normalized energy equation Is:

, _T* _*-V*T*) 2 Re

p*Cp (2 Re =_-_ ÷ Remax =

+ V*'(k*V*T*) + Ee ¢*

Pr

where Ec = Um, m_2

Cp (T H - TL)

is the Eckert number.

EO _-_t* + Ec Rema x _*-V*p*

Similarity parameters I/d and AR

The similarity parameters are obtained from the normalization of the

streamwlse component (x,u) of the momentum equation. Geometric slmllarlty

ensures similarity in the other components (y and z), e.g., the £/d - ratio

in pipe flow ensures that the boundary layer growth in the hydrodynamic

developlng length is similar.

lll
The set of parameters relevant for heat transfer in oscillating flow

then consists of: E/d, Rema x, Re03, Pr, Ec and AR. This set contains two

parameters more than in steady flow. One of these parameters, however, is

redundant for slnusoidal flow; If the velocity is

um - Um,ma x sin (03t)

then the maximum displacement of an average fluid element is calculated by

integration over half the cycle

2 Xm,ma x -o]T/2um,max sin(03t) dt - 2 _w

By the definition of A R

AR . 2 xm:m_x = 2 Um_m_y Z • 03_

but Rem_x = Um,m_x d _ = 4 Um:max

Re 03 d 2 03 d

03

therefore 1 d Re_max

AR = 2 & Re

03

For other, non-sinusoidal fluid motion, the integration will yield a

different proportionality factor than I/2. In any case, AR is a function of

&/d, Rema x and Re03. Therefore, the set of similarity parameters for heat

transfer is reduced by one to

AR or (L/d), Rema x, Re03, Ec and Pr.

Experimental data will show whether AR or £/d is more appropriate in

\

correlating heat transfer results.

112..
Calculation of the Reynolds number for regenerator matrices

The hydraullc diameter for regenerator matrices in the Stlrllng engine

data base Is calculated based on the definition used by Kays and London

(1984, p. 8):

dh/_ - _ Ao/Ah

where Ac - flow cross-sectlonal area

Ah = total heat transfer area

- heat exchanger length

dh - ¢ x vold volume/heat transfer area

¢ = porosity

Figure B-I shows a cross-section of two layers of a woven-screen

regenerator, where s is the wire spacing (i.e., the inverse of the screen

pitch) and dw Is the wire diameter.

$

CROSS-SECTION

Figure B-I :

S

FRONT VIEW (INFLOW DIRECTION)

Schematic of a wire mesh regenerator screen.

To a first approximation, the control volume (c.v.) contains a solid

volume

V s - _ dw 2s

If3
The porosity is defined as the ratio of void volume Vv to total volume Vt.

¢ . W . Vt - Vv

Vt Vt

The heat transfer area in the control volume is

Ah - _dws

Then the hydraulic diameter ls (Kays and London 1984, p. 8)

Ah Ah Ah 1

Thls is the expression commonly used for the hydraulic diameter.

The characteristic velocity in the Reynolds number Is calculated as the

average velocity in a cross-sectlon with effective flow area

Ac - ¢ s'

Then the characteristic velocity is

where V is the superficial mean velocity, i.e., the mean velocity of the

flow Just before entering or after leaving the matrix.

UmA c - VAf

where Af is the total regenerator frontal area, i.e., the cross-sectional

flow area when the matrix is removed.

114
APPENDIX C: VELOCITY PROFILES IN LAMINAR FLOW

The plots below show velocity profiles for laminar, incompressible,

fully developed, oscillating flow for eight dimensionless frequencies. The

velocity profiles were calculated from the equations given by Uchida (1956)

for pulsatile flow by taking the limit as the mean velocity goes to zero.

115
Re

,]

=i Re '= 3

..-- ,--, _, ,--

(LOCAL RRDIUS)I(PIPE I_DII.JS)

Re - 3O

.

|,

,,m

.

i"

._. .--' _, ,_. _ ..=- .-- ..,

(LOCAL B/_IUS)/(P].PE I_OIU5) (LOCRL RRDIUS)I(PIPE RADIUS)

Figure C-l: Velocity profiles in laminar flow

]]6 ORIGINAL PAGE IS

OF POOR QUALIT.Z
ORIGINAL PAGE IS

OF, POOR QUALITy

Re

!" I"

I"

i"

0

_t

uJ

N

I"

b

I

Re - i000 _"

% /

(LOCP,L _IUS)/(PIPE 1_SDIUS)

I"

0

i,!

l'

'T

p.

v

I, ._,

(LOCIK. IVIDIUS)/IPIP£ I_P.DIUS)

°-

Re

= 300

ffi 3000

Figure C-l: Velocity profiles in laminar flow

(continued)

117
APPENDIX D: EQUATIONS OF OBSERVATIONS OF TRANSITION

This appendix provides the equations describing the lines in Figure 3-8.

The ranges of applicability are based on the range over which the original

references provide experimental data to support these equations.

Reference

Grassmann and Tuma (1979)

Iguchi et al. (1982)

laminar _ transitional

transitional _ turbulent

Ohmi et al. (1982)

Park and Baird (1970)

Sergeev (1966)

Equation

Remax = 141 Rem0.75

Rema x = 15300

Rema x = 400R¢_e_-_

Rema x = 800 Rvr_j

Rema x = 800 Re_

Remax = 188 Rem2/3

Rema x = 700 Re_--_

Range

42 $ Re= _ 520

520 < Rem $ 943

100 < Re= < 800

100 < Reu < 800

18.4 _ Rem _ 546

35 _ Reu _ 1000

16 _ Re_ $ 1600

118
APPENDIX E: EFFECTS OF PRESSURE PROPAGATION

Resonance

A necessary condition for resonance In Stlrling engines Is that the

flew length between plstons Is greater than half the wavelength (_) of

pressure propagation.

a

f

where a - speed of sound

f = engine frequency

The smallest _ is obtained for small a (low temperature) and large f. For

the only air-charged engine in'the data base (Genset), the smallest _ is:

_1._1X 286.7 X 367 m/see

= f = 60 sec -_ = 6.42m

For He and H2-charged engines the worst case would be the He-charged SPDE-D,

the smallest _ Is:

f

/1.66 x 2079 x 350 m/see 10.5m

= 105 see -_ =

The flow length between pistons may be found in the literature for the 4L23

(Martini 1982, p. 32) as 1.4m. It can be estimated for the GPU3 from Urieli

and Berchewltz (198_, p. 36-37) as - O._m and for the UK engine from Dunn et

al. (1982, p. 68) as > 0.78m. For all other engines in the data base it Is

expected to be less than 3m, estimated from the exterior dimensions of the

engines.

I]9
The half-wavelengths, AI2, for both cases exceed the flow length

between pistons, therefore, resonance is not expected.

Shock incipience

A shock forms when pressure waves interfere constructively. This may

be predicted by the method of characteristics. Consider a piston that

oscillates sinusoldally In a tube (Fig. E-I).

Figure E-I: Oscillating piston in a tube

As the piston moves to the right, it continuously sends pressure waves to

the open end. These can be represented in the t-x-plane as shown in Fig.

E-2. Two pressure waves that were emitted by the piston at different times

may interfere if the one emitted later travels faster (smaller slope of the

t vs. x llne).

120
_t

IT

÷ sinusoldal piston motion I

I pressure _fc_p1:_ _o_.t_o_a_e___ _,

//_/charlcteristic of first pressure wave

ram.

Figure E-2: Method of characteristics

121
The lines descrlbing the pressure wave propagation are characteristics. If

the piston velocity is small compared to the velocity of sound, the slopes

of both lines will be nearly equal:

dt 1

(_)I = ;

dt = 1

(3"_)zz up ; a

dt dt

If a ) Up, then (_)I = (_)II

Then the two lines will intersect far from the piston.

In an englne, the maximum piston velocity is smaller than the fluid

velocity in the heat exchangers because the flow area is less than the

piston area. Therefore, a worst case is obtained if the piston velocity is

set to the highest fluid velocity found in the data base. The

characteristics In Flg. E-2 are most likely to intersect if the period of

the sinusoidal piston motion is small, i.e., the worst case should have the

highest frequency. Due to the temperature gradient in an engine, the speed

of sound varies. For the present simplistic analysis, the temperature is

assumed to vary linearly with x.

The analysis is one-dimenslonal and ignores contractions, expansions,

bends and regenerator matrices in the flow path. A fictitious worst case is

composed of all the most favorable conditions for shock incipience

encountered In the data base:

piston velocity amplitude

frequency of oscillation

engine flow length

Hma x = 0.15 UK-heater

w = 105 H z SPDE-D

L - 1.5m

L > 1.39m _L23

122
The temperature was assumed to vary linearly with x from 300K to I050K.

Consequently, the speed of sound varies as:

a(x) - /'_

-/'_/L [(Th - Tc)x+ To ]

Then the two characteristics are described by

2 [_T. - To)_' • To,-- n--_L] t#I = Th - To

t#ii _ to = _j_L'- 2 [I(T h - Tc)x + TcL

h To

_" UpQ I] - In_'/(Th - Tc)X + TeL + Upo)}]

where Upo is the piston velocity at to, when the pressure wave leaves the

piston and # denotes shock incipience due to interference of pressure waves.

Test cases for air, hydrogen and hellum were calculated for different

values of to but shock incipience was not observed in any of these test

cases (e.g., Fig. E-3).

123
TEST CRSE WITH HYDROGEN

i_

oX _-I

m

I--

I I I

._loo .04)0 1.2oo

DISTflNCE / m

I I

! .040 2.000

Figure E-3: Characteristics for a test case with hydrogen

124
APPENDIX F: AUGMENTATION OF AXIAL TRANSPORT

with

From-Watson (1983, p. 241), the augmentation coefffielent is

Au - T (We, Pr) VT= We - _Re_

T m

B=(Wo)

I - B=(Wo JPr)

87=(I - Pr*/=) B,(Wo)

where

We B"(Wo) + B'(Wo)

B=(Wo) - B'(Wo)

and

Wo=B(Wo) ÷ B'(Wo) - WoB"(Wo) - Wo=B"'(Wo)

B,(Wo) - WoWB,(Wo)

dn

Sn - _ (ber=(Wo) ÷ beiS(Wo))

Bet and bel are the Kelvin functions. A series solution for B and its

derivations is provided by Watson (1983, p. 241).

Based on this series solution, the plot of the augmentation function ¥

shown below was generated (Figure FI). The axial heat transfer can be found

as discussed In section 4.2 if an axial temperature gradient is assumed.

Sample calculations are not included here because they involve proprietary

geometry data.

125
| |

, i I , | I i " l ! | ] I , |, i e

Pr= 1.0

0.8

' ' I ''''I

I I0 I0' 103

Figure F-l: Augmentation function for axial heat transfer

1 2 6 _.$. GOVERNMENT _INTING nFFICE: 1988-5q_-157/504_2

NllllOnlll Aeronautics anU

Space AdrniniSlrat_*l

1. Report No.

NASA CR-182108

4. Title and Subtitle

A Survey of Oscillating

Engine Heat Exchangers

Report Documentation Page

2. Government Accession No. 3. Recipient's Catalog No.

Flow in St|fling

7. Author(s)

Terrence W. Simon and Jorge R. Seume

9. Pedorming Organization Name and Address

University of Minnesota

School of Mechanical and Aerospace Engineering

Ill Church Street S.E.

Minneapolis, Minnesota 55455

Administration

12. Spon_ring Agency Name and Address

National Aeronautics and Space

Lewis Research Center

Cleveland, Ohlo 44135-3191

5. Report Date

March 1988

6. Performing Organization Code

8. Pe#orming Organization Report No.

None

10. Work Unit No.

586-01-11

11. Contract or Grant No.

NAG 3-598

13. Type of Report and Period Covered

Contractor Report

Annual

14. Spon_ring Agency Code

15. Supplementa_ Notes

Project Manager, James E.

Research Center.

Dudenhoefer, Power Technology Division, NASA Lewis

16. Abstract

Similarity parameters for characterlzing the effect of flow oscillation on wall

shear stress, viscous dissipation, pressure drop and heat-transfer rates are proposed. They are based on physical arguments and are derived by normalizing the

governing equations. The literature on oscillating duct flows, regenerator

and porous media flows is surveyed. The operating characteristics of the heat

exchangers of eleven Stirling engines are described in terms of the similarity

parameters. Previous experimental and analytical results are discussed in terms

of these parameters and used to estimate the nature of the oscillating flow under

engine operating conditions. The operating points for many of the modern Stirling engines are in or near the laminar-to-turbulent transition region. In several engines, working fluid does not pass entirely through heat exchangers during

a cycle. Questions that need to be addressed by further research are identifled.

17. Key Words (Suggested by Author(s))

Stirling

Oscillatlng flow

Flow

Heat exchangers

18. Distribution Statement

Unclassified - Un|imited

Subject Category 34

19 Security Classif (of this repom) 20 Security Classif (of this page) 21 No of pages 22 Price"

Unclassified Unclasslfled 132 A07

NASA FORM 1626 OCT 86 "For sale by the National Technical Information Service, Springfield, V_rginia 22161