NASA / TM--2002-208914

Visualization of Flows in

Beds of Twisted Tapes

R.C. }--tendricks

Glem_ Research Center, Cleveland, Ohio

M.J. Bram_ and D. Peloso

University of Akron, Akron, Ohio

M.M. Athavale

CFD Research Corporation, Huntsville, Alabama

R.L. Mullen

Case Western Reserve University, Cleveland, Ohio

Packed

November 2002
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NASA/TM--2002-20891_4

Visualization of Flows in

Beds of Twisted Tapes

R.C. Hendricks

GleAm Research Center, Cleveland, Ohio

M.J. Braun and D. Peloso

U_liversity of Akron, Akron, Ohio

M.M. Athavale

CFD Research Corporation, Huntsville, Alabama

R.L. Mullen

Case Western Reserve University5 Cleveland, Ohio

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Prepared for the

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sponsored by the Pacific Center of Thermal-Fluids Engh_eering

Honolulu, Hawaii, May 16-1!), 1!)!)!)

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Available electronically at http://gltrs.zrc.nasa.gov
VISUALIZATION OF FLOWS IN PACKED BEDS OF TWISTED TAPES

R.C. ttendricks

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

MJ. Braun and D. Peloso

University of Akron

Akron, Ohio 44325

M.M. Athavale

CFD Research Corporation

Huntsville, Alabama 35805

R.L. Mullen

Case Western Reserve University

Cleveland, Ohio 44106

INTRODUCfION

Although similar visuNization materials _ld techniques have been discussed in previous work by ttendricks

et al. (1997), we realized that printed frames, grabbed from videotapes of the flows in beds of packed spheres, failed

to reproduce what was visualized. Herein we describe the events associated with packed beds of twisted tapes, but

to fully appreciate the complexity of tile flow fields, it becomes necessary to watch tile videotape recording, flowever, tile Full Flow Field Tracking (FFFT) method (Braun et al., 1988) can be applied to visualize and quantize the

flow patterns and fluid velocities within a packed bed or a porous medium.

Packed beds of twisted tapes may serve as an alternative to porous-media packed beds in heat pipe applications

in low- mid high-body *brce fields, such as in space and gas turbine applications. Potentially, twisted tapes could

also function as reaction surfaces where uniformity of mixing is sought. In this study the test section assernbly

simulated a canister of twisted tapes for heat, mass, and reaction exchange. An assembled cylinder would contain 6

to 10 such canisters in series. In turn, several cylinders would be bundled into an array. We tested only one simulated canister, although for m_y practical application simulation of an entire cylinder and array would be necessary.

Power developed along the path and heat transfer and pressure drops downstream would modify the *lows in the

upstream leading canister even to the point of choking the element. Choking is quite serious because the power

generation in a practical system is nearly constant and failure of"the element becomes imminent.

ANALYSIS

Using twisted tapes (fig. 1, from Smithberg and I,andis, 1964) is a well-known method for augmenting heat

transfer in tubes at the expense of pressure drop in single-phase flows (Hong and Bergles, 1976; Lopina and Bergles,

1969; Bergles, 1998). Yet at constanl pump power, twisted-tape, swirl-flow heat transfer can be increased by 20%

over that of a straight tube (Lopina and Bergles, 1969). We will use these references as the starting point for

representing the ideal packed bed of N twisted tapes (cf figs. 1 and 2) and work toward a porous-media model. The

details are presented in appendix A.

For a single twisted tape in a tube the tangential fluid velocily is usually assumed to be linear with radial

position (rotating slug flow) or

2_Ur

vo - (1)

NASA*I'M--2002-208914 1
_UD

Yt--

I4o

(2)

where r is the radial position, D is the twisted-tape diameter, H(, is the twist through 360 ° (one full wave ), and U is

the bulk average axial velocity. (All symbols are defined in appendix B.)

W

where W is the mass flow rate, p is the average fluid bulk density, t is the tape thickness, and w is the tape width, and

for f << D, wt --9 Dr.

Single Twisted Tape in Tube

From figure 2 of Smithberg and Landis (1964), the data closely follow equation (1), except at the wall and

centerline where % = 0, and nem'ly fit the following form:

U Co1-_ L

(4)

which is a close approximation to equation (1). For these data, 1Re D = 137 000, HolD = 3.62, D = 3.51 cm

(1.382 in.), and g0,ex p = 71.63 m/s (235 ft/s) at r = 1.65 cm (0.65 in.). At that point r/r o = 0.94, v0 = rc(2r/D)(D/Ho)U,

or U = 82.3 ln/s (270 ft/s). From equation (4)

235

Co= 27-----_0= 1.14

0.763

(5)

Here v 0 --->U and even ff)r this case neither an effective velocity (eq. (6) wilh CO= Ce) nor an effective flow path

(eq. (Al 2b)) is sufficient lo account lbr the measured pressure (h'op increase in terms of..fl£).

For laminar flows with Reynolds numbers less than 150, Date (1974) determined that DIH o corrections are not

required. However, for 150 < Re < 2000 (and for turbulent flows), D/H o corrections are necessary. Dora the fifo data

of Smithberg and Landis (1964), Q, --->3 with an equivalent velocity defined by using equation (2),

<

_= 1+ =L _" H°)J

(6)

The simple empirical form (eq. (6)) tends to group the turbulent friction data of Smithberg and Landis (1964)

at a higher Reynolds number and identifies (D/tto)2 as a significant parameter for the analysis, ttowever, there are

additional Reynolds number and surface roughness dependencies (Gambill and Bundy, 1962). For example, at

1The data point ReD = 137 000 with D/DH = 1.716 becomes ReD_¢ = 79 837. Extrapolating the HiD = 3.62 locus may give

fljl _= 2.8, and this ratio may also be affected by roughness.

NAS A/TM------.2002-208914 2
Re = 25 000 the agreement of friction factors is good at high Ho/l_), but at Re = 6000 the agreement is better at low

Ho/D bul is still nol that good. The surface roughness is nol given and is assumed lo be that of a commercial tube.

Gambill and Bundy (1962) correlate lhe isothermal, single-twisted-tape data of several investigalors with

different surface roughness 8/D e factors.

( 0.21 ]( Re e _-n

(fs-fa)eiso=t Jt (A43)

where

o,,exp[ (A43a)

=4So=t-2g )t o. nu2)

and y is the number of tube diameters per 180 ° of twist 0' = H/D or 2y = HolD ).

(A43b)

Ho/D = 2y

6000

22 1.12

10.3 1.31

4.34 1.97

3.62 2.23

81D_ = 0

I 25000I

1.05

1.13

1.41

1.51

8/D_ = 0.00005 8/D_ = 0.0005

80000 6000125000180000 6000125000180000

f, = 4/'o = 4(0.046/Re °'2) :£ from equation (A43)

1.02 1.12 1.06 1.03 1.19 1.16 1.13

1.06 1.34 1.15 1.08 1.52 1.42 1.36

1.20 2.04 1.48 1.25 2.61 2.31 2.10

1.25 2.32 1.61 1.32 3.04 2.66 2.40

Packed Bed of Twisted Tapes

In the packed-bed experiment described herein the tapes were not bounded by tube walls. They were assembled

into a uniform matrix of twisted tapes with the same twist direction, twist-to-diameter ratio HID o, and thickness t.

The boundary conditions changed from those of a single twisted tape. The tangential velocity became zero at the

center of the twisted tape and at the tangent points of the packed bed of virtual tubes of diameter D Obecause the

velocity fields were counterrotating. The rotating velocity external to the confines of the virtual cylinders was

assumed to be small, thus permitting a local region of axial flow. As noted earlier, corrections for H/D o effects in

low-Reynolds-number flows are not required (Date, 1974). For HID o > 3.6 and w ----_D, the tape lengths before and

after the twist do not differ substanti ally.

In packed beds the superficial velocity is related to the bed porosity by

g o

u : -- (7)

13

where U o is the empty or unpacked bed velocity and _ is related to the bed volume V as

gsolid

Vtotal

(8)

NASAfI'M--2002-208914 3
Forthepresent caseachof48twistedtapeswasconsideredasencasedinavirtualtube,wherethetapewidlh

w = 0.3234 cm (0.1273 in.) and the tape thickness t = 0.1275 cm (0.0502 in.).

Do=(W 2 +12) °'5 (A4)

Thus, D o = 0.348 cm (0.137 in.) and H o = 5.503 cm (2.17 in.). For this geometry tIJD o = 15.8 ---> H/D o = 7.9, or

three 360 ° twists in 16.5 cm (6.5 ill.).

For the packed bed of 48 twisted tapes considered herein, the orientation of the twist angle was somewhat

irregular (figs. 2(b), (d), (e), and (f)); however, by using tile prior relation for porosity and variations in t and w, the

estimated porosity range becomes (see measurements and estimates in appendix A)

with a mass flow ran,;e of

where A o is the cross section of the empty tube.

0.59 < e < 0.63

1.6 < upA° < 1.7

W

(9)

(10)

Parameters tbr Single Twisted Tape in Tube

The analysis and data of Smithberg and Landis (1964) were considered to provide insights into the effects of

twist H/D or H/D o mid pressure drop. Refbrmulating the Ergun parameter YErgun (Ergun, 1952) in terms of" the

Faming friction factor (see appendix A) gives

where

1141

fSmifiaberg-Laxldis = _ Ergun

YErgun= [ jtl_ j tLo)

(A14)

(A16)

G O = pUo (A15)

At high Reynolds numbers

YErgun --->Constant --+ 0.014 for e --+ 1 and Re >> 2000 (A25)

where 0.014 is the coIranercial rough-tube equiv_ent. In equation (A16), L o represents the straight-line distance

between pressure taps. The twisted-tape length L varies little from L o over the practical range of tapes. For laminar

flows the correlated pressure drops are weakly dependent on twist H/D or ttJD, yet for turbulent flows the pressure

drop data are strongly dependent on twist. In most cases dp/dz -9 AP/L tbr single tapes in tubes. So in terms of

Reynolds number the packed-bed Reynolds parameter XErgun becomes

where

2el GoDp ]=2e X

eesmithberg-Landis = "_-[_(1--7"7"_J V Ergun

(A17)

NAS A/TM------.2002- 208914 4
XErgun -- (1 -- e)bt

2e

(f Re)smithberg -La"dis= 7- YErgunXErgun

and the generalized relation carl be expressed as

(A18)

(A19)

XErgun YErgun =70.9+Ig2(H,ks)]XErgun (A83)

From extrapolating the data g2(tt/L), k s) --> 0.014 mid represents a lower bound of the data as illustrated in figure 3

and labeled "single twisted tape." To illustrate the dependency of tile turbulent flow data on t_;/D, we normalized

the Ergun friction factor by using equation (6). The modified Ergun friction factor is shown in figure 4 as

YErgun) modified = YErgun 2 n0.5 (11)

and tends to follow the simplified form

XErgunYErgun = 70.9 + 0.014XErgun (A83a)

TEST FACIMTY

The test facility consisted of an oil tunnel, flow system components, video equipment, a laser, lens systems, data

recorders, and a test configuration (fig. 5). The test section consisted of 48 twisted (spiral) Lucite tapes assembled

into a bundle and placed into a clear Lucite tube 2.54 cm (1 in.) in diameter. The tape bundle simulated, for

example, a system of catalytic reactive surfaces, heat transfer augmentation surfaces, or flows through strata of

porous media. The twisted tapes were made t'rom clear, polished Lucite sheet 0.152 cm (0.06 in.) thick cut into strips

0.318 cm (0.125 in.) wide by 17.8 cm (7 in.) long (nominal dimensions; measurements are discussed in appendix A).

These strips were gripped 0.635 cm (1/4 in.) from each end and twisted with three complete twists (figs. 2(a), (d),

and (f)). Figure 2(b) is a cross section of the actual packed bed, and figure 2(c) illustrates an ideal packed bed. A

flow screen with 0.08-cm- (0.032-in.-) square mesh made of 0.023-cm- (0D09-in.-) diameter wire was placed across

the tube inlet and attached to a square support (figs. 2(d) and (e)). For Borda inlet flows the screen restrained the

axial movement of the twisted tapes and served as a screened orifice inlet when the test section was reversed. In the

latter case, the twisted tapes were restrained 0.95 cm (0.375 in.) downstream by the visualization rnirror. Some

movement of the packed bed was noted. The movement would slightly disturb the inlet and exit flows but was not

expected to alter the developed flow field.

The assembly (fig. 2((1)) was then placed into a closed-cycle oil tunnel (fig. 5) with a square support fabricated

to retain the tube in the flow field and block the remaining tunnel cross section (152.4 cm by 7.62 cm by 7.62 cm; 60

in. by 3 in. by 3 in.). Both the upper wall and the viewing port walls of the tunnel were Lucite (fig. 5). The index of

refraction of the oil matched that of the Lucite, and magnesium oxide particles were used as flow tracers. The flow

field was visualized by using the Full Flow Field Tracking (FFFT) method (Braun et al., 1988). Laser light sheets

illuminated two-dimensional sections of the tunnel along the flow path. Transverse visualization was accomplished

by placing a rnirror in the tunnel downstream of the 4g-twisted-tape bundle and at approxilnately 45 ° to the flow

axis. A second mirror placed above the tunnel (not shown in fig. 5) projected the view to the television camera.

Cross tunnel traverses were accomplished by small rotations of the mirrors. These traverses provided insights into

the three-dimensional nature of the flow field.

NASA/I'M--2002-208914 5
Tilecoherent-beam, continuous-wave,rgon-ionlaserwasdirectedbymicrometricadjustablemirrorsthrough

twocylin(kical lensespositionedat90° to each other and through tile Lucite tunnel window and into the test section.

Tile lighl sheet was approximately 0.01 cm (0.004 in.) thick, and the flow was seeded with nlagnesium oxide flow

tracers. Micrometric adjustments controlled scarming of the light slices across the test section and provided a threedimensional visualization of the flow field, which was videotaped at 30 Hz.

EXPERIMENTAL RESULTS

Visualization

Figure 6 represents the axial flow field along the centerline of the packed bed of twisted tapes. Flows across the

inlet were markedly influenced by the twist and packing of the bed. At the inlet the vena contracta normally found at

the inlet of an open tube was limited to a minor region near the tube wall entrance (fig. 6(a)). Beyond this entrance

region the flow was rapidly entrained into minor spiral perturbations (HiD o = 15.8) within less than 0.1D. However, for flows near adjacent solid boundaries there is clear evidence of local spiral flows within the field of focus.

Flows in the boundary layer of each twisted tape tended to spiral as expected (fig. 6(b)). Thus, the flows tended to

follow typical boundary layer flow patterns: no slip at the surface and a region of viscous flow closely aligned with

the surface topology blending into a region of fully developed flows with minor perturbalion of the streamlines. As

the pressure drop was increase(], these regions became less distinct in thai: the boundary layers became thin and

difficult to visualize. There still persisted a region perturbed by the spiral nature of the surface, but it became less

distinct as the flow velocity increased.

By aligning the laser sheet transverse to the flow we obtained a circular cross section of the flow. The packed

be(] of twisted tapes appeared as small, randomly oriented, rectangular blocks in tile flow field (fig. 2(b)). There

appeared regions where the flows were jeU:ing between adjacent blocks, regions where the flows were predominantly clockwise or counterclockwise, regions where the flows were upward or downward, and wall boundary layer

flows at both the "bh)ck" surfaces ail(] the interfaces between the tube and the twisted tapes (fig. 7). Notably absent

were local vortex flows, with the possible exception of one configuration. Although jeU:ing should produce vortices,

the three-dirnensionality of the flow may have precluded observation. It appeared that engendered vorticity was

captured within tile axi_ spiral flow field, although not readily visualized.

Packed Bed of Twisted Tapes

The pressure drop and flow data parameters in table 1 were plotted in figure 8. The errors in the difference ill

static pressures upstream mid downstrealn increased at lower Reynolds numbers, and those data should be considered suspect. It is evident that the Borda inlet configuration with a screen had a slightly higher flow resistance than

the orifice configuration with a screen and that the orifice configuration without a screen had the lowest resistance.

Consistent sets of parallel lines could be passed through each set of data in figure 8. Shown for reference are tile

Ergun (1952) porous-media model and tile Date (1974) single-twisted-tape Inodel. Ill general, the data were lower

than the Ergun model over the range of data taken. The average fluid temperature was 22 _+ 1 °C, the viscosity was

0.91 poise, aa_d the density was 0.9 g/cm _. A suggested general form for N twisted tapes in a tube, following

equation (A83), is

XErgunYErgun = 45 + 0.009XErgun (A85a)

which is considerably below that suggested for flows in porous media

XErgungErgun = 150 + 1.75XErgun (A84)

suggesting significantly less flow resistance for the packed bed of twisted tapes.

For flows through packed fibrous beds the resistance for parallel fibers is about one-third that for perpendicular

fibers

NAS A/TM-------2002- 208914 6
1

( un un)pa allel = =¢ un Un)pe pend cul (A99)

These expressions agree with those presented by Hersh and Walker (1980), but an average of the two forms more

closely approximates the data of Sullivan (1941 )

46.6 + 58.8

____XFrgungFrgun-- -- 52.7 (1109) 2

and is similar to the reported (laminar) data (table 1 and eq. (A85)).

Single Twisted 'rape

Parameters X[!r,,u n and YFrgun from the selected interpolated data of Date (1974), Koch (1958), Gambill and

Bundy (1962), and _mithberg" gnd Landis (1964) are plotted in figure 3 for the range 056 < Ho/D < _. The dependency on Ho/D is evident mad is bound by the porous-flow loci (eq. (A84)) to that of a single twisted tape at high

Reynolds numbers (eq. (A83)). This conclusion is based on the data of Koch (1958), which were tN_en from figure 7

of Smithberg and Landis (1964).

The results of Sparrow and ttaji-Sheikh (1966) for laminar flows can be expressed as (see page 17)

YErgunXErgun = 70.94 (A39a)

Also plotted in figure 3 are the data for 48 twisted tapes in a tube _ong with equation (A84).

Figures 9 and 10 represent the loci of 48 twisted tapes in a tube, a single twisted tape in a tube, porous-media

flows, and interpolated turbulent flow data for a single twisted tape in a tube corrected for swirl velocity.

VIDEOTAPE RECORDING

The complexity of the flow field, whether virtual or experimental, became vivid through the videotape recording, which is included as .avi and .mov files on the supplement CD in the printed version of this report. Visualization of the flow field reveals flow threads, wakes, stagnation zones, and the influence of the twisted-tape interfaces.

The flow threads can be observed during a scan of the flow field from the front to rear lateral walls. These flow

threads weave through the packed array of twisted tapes in the bulk flow direction. Details of the flow boundary

layer close to the wall and progressing through the packed bed to the opposite wall are revealed. The video can "also

be used to determine quantitative experimental information, such as the flow velocities, by using the FFFT technique (Braun et all., 1988).

Click here to play movie

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

NASA/FM--2002-208914 7
SUMMARYOFRESIJLTS

Theflowexperiment consistedofthreeprincipal eements: anoilrunnel 7.6cmby7.6cm(3in.by3in.)in

crossection, acylindrical robecontainingthepackedbedoftwistedtapesinanarbitraryarray,andaflowcharacterizationmethodology, FullFlowFieldTracking(FFFT).Theindicesofrefractionoftheoilandthetestmatrixof

twistedtapeswerecloselymatched, andtheflowwasseededwithmagnesiumoxkieparticles. Planarlaserlight

providedatwo-dirnensional projectionoftheflowfield,andatraversesimulatedathree-dimensional imageofthe

entireflowfield.Flowswereobservedneartheinletofthecylindrical tubehousingthebundledarrayoftwisted

tapes,aItheinterfacebetweenthetubewallandthetwistedtapes,andwithinthebundleoftwistedtapes.

Theflowfieldwasthree-dimensk)nal andmostcomplextodescribe. Themostproirfinent findingwasflow

threads. Theaxialflowappearedtospir_alongthetwistedtapeswithintheconfinesofavirtu_distortedcylindrical

boundary. Theflowfieldappearedtobesimulatedbyapackedarrayofverythinvirtu_cylinders, withthe

exceptionofthespirNeffectduetothetwist.Theeffectsofrandompackingandbedvoidscreatedvorticesand

disruptedthelaminarflowbutminimizedtheentranceeffectsoftheunpackedtube.

Theresultsofseveral investigators forflowsingeometries withasingletwistedtapewerean_yzed. These

resultsarerelatedtotheErgunmodel inappendixA.(SymbolsaredefinedinappendixB,andacomprehensive data

tableisgivenintable1.)Thesingle-twisted-tape resultsofSmithbergandLandis(1964)havebeenusedtoguide

theanalysis. Thedatafor48twistedtapesinatubewerecorrelatedbyusingtheErgunmodelfbrflowsinporous

media.Thepressuredropandflowdatafbrthethreegeometricconfigurations (Bordandorificeinletswith

downstreamrestrainingscreenandorificeinletwithoutscreen) havedistinctflowcharacteristics differingupto

('re 13 J< The averages for the combined data sets were lower than the Ergun model by a factor of 3 for the packed bed

of 48 twisted tapes in a tube and by a factor of nearly 1.6 fbr a single twisted tape in a tube. These results suggest a

lower flow resistance for a packed bed of 48 twisted tapes in a tube than for either porous-lnedia flows or singletwisted-tape flows. Further investigations including different geometric configurations and computational fluid

dynamics analysis are suggested.

NAS A/TM-------2002- 208914 8
APPENDIXA

RESIJLTSOFSEVERALINVESTIGATORS SCAI,EDTOERGUNMODEL

Becausewereallydonotknowhowtocorrelatehedatabetweenasingletwistedtapeandmultipletwisted

tapesinacylinderorlube(figs.1and2),wewillbeginwithsomefamiliardefinitionsoffrictionfactorand

ReynoldsnumbersandarriveattheformsimilartothatdevelopedbyErgunforporous-media flows.

AP=(4f'](pu2LI (general Fanningfrictionfactor)

t. 2 _. D,v )

(A1)

Re- pUDH

and for now, let p = Constant and Cf = 4_ as used, for example, by Sparrow and H@-Sheikh (1966), Hong and

Bergles (1976), and Gambill and Bundy (1962).

Now lel us define some packed-bed paramelers. Lel the average or superficial velocity within a packed bed of

one or more twisted tapes be

U--

U _ Uo

g g

(A2)

where U is the velocity in the tube without tapes. The bed porosity is defined as

Ntw

D 2

g--

4

(A3)

where N is the number of twisted tapes, t their thickness, w their width, and Do the virtual twisted-tape diameter

(fig. 2(c)).

DO =(w2+ t2) °'5 (A4)

We can now define the characteristic length of the packed bed in terms of sphere diameter. Note that D H =

4A/5, where A is the cross section of the flow area and S the wetted perimeter. For a tube DH = Drab e and for a

uniform bed of spheres 6V/A s = 6/a_, = Dv ----->Dsphere (Bird et all., 1960). This factor of 6:4 or 3/2 will become a

sc_ing parameter for the Ergun relation. The characteristic length of the packed bed is

D H _ Rh = e_. (A5)

4 a

where R h is the ratio of tile bed cross section available for flow to the wetted perimeter, which is equal to the ratio of

the volume available for flow to tile total wetted volume and is equal to the bed porosity divided by the ratio of tile

wetted surface to the bed volume. The specific surface area is

a 6

av - - (A6)

1-e Dp

NASAfI'M--2002-208914 9
where av is equal lo tile ratio of tile total sphere surface lo its w)lunle and is equal to the ratio of the weUed surface to

the solid volume.

Combining equations (A5) and (A6) gives tile characteristic length of the packed bed as

_ Dpe DH - Rh (A7)

4 6(1- e)

In terms of one or more of N twisted tapes continued within a cylinder or tube of diameter D, at_d with equation

(A3),

4/ ¢2/

m _ Ntw

DH = _ De

gD + 2N(t + w) 1+ 2N(t + w) (A8)

_D

When equation (A8) is substituted into equation (A7), Dp becomes

3D (l-e)

Dp - 2N(t + w) (A9)

1+

7rD

Note that in figure 2(c) the dark shaded areas A, B, and C are within the bounds defined by the twisted-tape

width w and thickness t, the tri-circular loci with radii of D J2, and tile triangle with vertices 1,2,3. The area

B + _ + _ = wt / 4, the area C - c + b = wt / 4, and the area A ----a ---b = 0. Therefore, the dark shaded area (solid area

of tile tape) within the lriangle is wf/2. Conlinuing with lrian.gle 4-2-1, C 1 - c 1+ b1 = wt / 4, B1 + Cl + a = wt / 4,

and A m- b I - _ = 0. Similarly, for triangle 2-5-3, C 2 - _ + b2 = wt / 4, B2 + c + _2 = wt / 4, and A 2 -b 2 - a2 = 0.

Summing these areas gives the total solid twisted-tape area within the confines of the hexagon. Extending these

results provides a generalized lk)rrn for N twisted tapes within the confines of a hexagonal space that approximates

that of a circumscribed cylinder or tube. This generalized forln is an ideal model for packed beds that is expressed

only in terms of tape width and thickness.

t W

22 2 tw

em°del -- 1 D _D - 1 _ t2 + w2 (A10)

_ o 2 o

lm2 mm

and for w = 2t (see also the section Some Sample Calculations al the end of tills appendix)

e= 1-_= 0.538

Compm'ing the model to the experiment where 48 twisted tapes with average width and thickness {w) = 0.3234 cm

(0.1273 in.) and (t) = 0.1275 cm (0.0502 in.) were contained in a 2.54-cm- (1.0-in.-) diameter tube gives all average

porosity, from equation (A3),

@exp) = 1-3 = 1- 192(0.3234)(0.1275) = 0.61 (All)

27r 7r(2.54) 2

NAS A/TM-------2002-208914 l 0
which indicates that the experimental test tube was well packed. For a single twisted tape in a tube (Smithberg and

Landis, 1964), where t = 0.0559 cm (0.022 in.), w --->D o = 3.51 cm (1.382 in.),

1 4t = 0.9797 (A12)

gSmithberg-La_dis = - _D---"7

A probleln arises because this lbrmulation is independenl of twist HID. From structural mechanics tile twisting

of tapes is considered to follow linear mechanics with little change in length, unless plastic deformation with

subsequent annealing takes place. As a result of this assumption the bed porosity is not a function of H. However,

the flow path is a flmction of twist H/D, and consequently, the characteristic length must be inodified to reflect H.

The spiral path is defined in terms of the pararneter (p as

x = rcos (p; y = rsin (p; Z = h(p (A12a)

and the path-length arnplification ratio becomes, for the number of 2rr twists _*t,

L _ 2gntr 1

zo zo t.2 <d ]

(A12b)

For very large tt/D (slow spiral), n t --> 1 and L/Z o --> 1. For very small tt/D (rapid spir_), t, t --> M >> 1 and L/Z o -->

2rcMr/Z o. This amplification is not strong enough to account for tile pressure drop increases due to the twist.

Models for Single-Twisted-Tape-in-Tube Analogy

Smithberg and I,andis (_]:964).--Smithberg and Lm_dis (1964) used the local[ average velocity superficial bed

velocity) in their correlations. Consequently, their friction factor mad Reynolds number relations can be scaled

directly in terms of the packed-bed parameters gErgm_ and XErgmv Substituting equations (A2), (A7), and (A1 1) into

equation (A1) gives the pressure drop parameter YErgun in terms of the Darcy friction factor as

4,+=4,+ n ,hbe -Land =t07gj2)t-U)=5t-Uo2)t _-77)t (A13)

1141

fSmithberg-Laxidis = "_ Ergun (A14)

where

G O = pU o (A15)

=t )t Lo)

(A16)

Here L o represents the straight-line distance between pressure taps. The twisted-tape length L waries little from L o

over the practical range of tapes, and for larninar flows tile correlated pressure drops are weakly dependent on twist

H/D. However, for turbulent flows the pressure drop data are strongly twist dependent. In most cases @/dz --+ M'/L

for single lapes in tubes. Therelbre, in lerms of Reynolds number the packed-bed Reynokis pararneler XErgua

becomes

NASAffM--2002-208914 11
where

eesmithberg-Landis =7-k(1--77757J7- Ergun

2E

(/ee)smithberg_Landi s - _ YErgunXErg un

(A17)

(A18)

(A19)

For turbulent flows the YErgunXErgun product is dependent on both Re and H/D. Smithberg aa:d Landis (1964)

provide a simple expression for the Fanning friction factor. It is important to recall that herein we used H/D = 18(I °

twist, whereas Smithberg and Landis used HiD = 36(I ° twist, where H and H o are measured _ong the axis parallel

to the tu be centerline, 2

where

H

H°sD = (-D)Smithberg_ Laaadi =2( H )

(A20)

004 + 1/ <2] e 

H -o 5

/]

(A21)

Adjusting the constant 2.1 and the exponent -1.2 slightly gives a better fit to the data, and the normalized ibrm

becomes

goH° =± I/ /l 1+105[---D--- 1) j Resmithberg_Laaadi s (122)

wherefo = 0.046Re 0.2. In terms of equation (A21), equation (A19) becomes

YErgunXErgun =_jT[go(_.____)](fo 9 H o ee)srmthberg_Landis _ 0.207[go(____]]Res}_ithberg_Laaadis g [ _-Ho )J 08 (123)

After substituting equation (A17), equation (A23) becomes

0.8 H 0.8

YErgunXErgun = 0.15E [g(-_)]Xl_rgun (A24)

o l' '

"Yet the number of 180 twists will be twice the number of 360 ° twists for a fixed tube length (i.e., ,z180° twists = 2n360° twists)'

and some authors use twist count rather than measured values of H and Ho.

NAS A/TM-------2002-208914 :l2
where

AtveryhighReynoldsnumberssurfaceroughness willpromoteeddyburstsneartilewallsandseparationeffectsas

theflowattemptstotrackthetwistedtape,implyingthat(figs.8mad10

YErgun --4 Constant ---> 0.014 for e --->1 and Re >> 2000 (A25)

where 0.014 is the commercial rough-tube equivMent (see eqs. (A54) and (A55)). For two-phase flows excessive

vapor generation, or holdup, would be at_ticipated.

tIong and Bergles £1976).--A similar set of sc_ing pm'ameters can be developed for the data of Hong at_d

Bergles (1976):

ReHong_Bergle_ - U - [---7-- --7 (A26)

From equation (A8) with N = 1 and (w + t) --> D,

_[ +2/+w"

D_ 1 _-

Dp _7_

(A27)

ReHong_Bergle s ---> 1.09 XErgun

g

(A28)

Cf, Hong_Bergle s ( _-__ 2](D]=2(0-_2 ](Dp](D--D--]E2 2 (l+2t+W] 4(1+ 2 ]

=_.O.5pu)_.L) _Go)_.LA.DF): 2YErgun -3 [ ./rdJ JY r un

where LJL-----> 1. With t = 0.046 cm, D o = ;1.02 cm, and w --> D o.

0.046

gHong_Bergle s = 1 0.943

1.02 7r

4

Equation (A10) would give 0.945, so w --->D o is a good approximation.

Cf,Hong_Bergle s = 2.314YErgun

For laminar flows in a half-tube configuration, equivalent to a tube with a single twisted tape,

(A29)

(A30)

(A31)

NASAfI'M--2002-208914 13
8/1+ /2

=e)=on=_=e,=,e= = "='=un=='=un = 183.6 (A32)

YErgunXErgun = 68.6 (A33)

We will later show the relation to the work of Sparrow and Hail-Sheikh (1966) as

2

(CfRel =183.6=( DH ] (C,,.Rel

\ /Hong-Bergles _. D ) _ a /Spaxrow Haji-Sheikh

(A34)

where DJD = 0.5682 is found 1}orn equation (A8) by using lhe geometry of Hong and Bergles (1976), where * =

0.046 cm, w = 0.97 cm --->D o, D = 1.02 cm, and (DII/D)2 ---> 0.334.

Date (1974).-------For tile work of Date (1974) tile scaling factors become, for (t + w) ---> D,

pud (PUoDpI[_I=2[I+2 I

0 [ J<un (A35)

l(dp]( D _ l(dp]( Dp 11D 11!;2 111+21

fDate = 7 _.7)_ 7) = 2"_.7)_ _)i-_'Tp ) = _"['_-JYErgun

(A36)

(fRe)Date =-_(--7--j gErgunXErgun

(A37)

For laminm" flow in the half-tube configuration, or for a tube with a single twisted tape, and &---->1 (see Weigand,

1948, and eq. (A76))

fDate ReDate = 42.19 (A38)

YErgunXErgun = 70.9 (A39)

which is in good agreement with equation (A33). ['or a = 0.93& YErgunXErgun = 61.8; and for a = 0.9797,

Yl__;rgunXErgun= 68.0.

For ReDa e < 150_ tape twist H/D has little influence on the Fanning friction factor. For 150 < ReDate < 1000_

fDate = fDate half-tube + AfDate

NAS A/TM-------2002-208914 14
_Re_Date=_ate half tubeReDate(l+ _ate _DateReDate half tube RDate/=421911_ _ _Date_Date_ 4219

1+ 2 1+ 2 2 = 1+AZDate/_--7-j 4-7_1_j_-/--7--j YErgunXErgun (140)

and for _ --> 1

YErgunXErgun = 70.9(1 + 0.026AfDateXErgun ) (141)

and for 150 < ReDate < 1000

I( D ) k.{D'O'IH I 1( ReDate ]

0 1 -- (A41 a)

AfDat e = 0.0881_ Date + " | [Date 1Ogl0 150 J

For Rel)at e > 1000, tile Smithberg and Landis (1964) expression, or modified expression, for the Fanning fi'iction

factor can be converted to fl)ate. Recall equation (A21):

fSmithberg-La_dis = 0.046 + 2.1 - 0.5 / Re -n (A21)

where

n = 0.211 + 1.7(--_-) -°'5 ]

and the conversion for ReDate > 1000 follows as

D 2 -1.2

(f Re)i)ate= (57-H){0.046+2.1[(-_--)i)ate-0.5] }Re_-m (142)

where

m=0.2 1+1.7 2

Date

The D/D H = 1.66 given by Date (:1974) shotlld be nearly 1.6366. Date gives no w_lues for tape width or thickness,

and estimates of e will give a range 61 < (gErgunXbs.gun)Date < 71 lk)r laminar flow in a half-tube or in a tube with a

single twisted tape. Also note that Date's calculations underpredict turbulent friction factor data by 30%.

Gambill and Bundy (1962) and Gambill et al. (1961) evaluate swirl-flow heat transfer along with isothermal

data on 1Yiction coefficients. The relation given

NASAfI'M--2002-208914 15
where

( 0.21 ]( Re e _-n

(fs- fa)eiso=tTrr jt ;

n= 0.81 exPI-1700(_e ] ]

(A43)

(A43a)

fa = 4 fo = t.---_-x )t O._pU2 )

(A43b)

 a4 c,

_D- 2t + 2D

and where iv is the number of tube diameters per 180 ° of twist (2n/)), D e the equivalent or hydraulic diameter (over

the practical range 5/4 < y < oo, D e ----->DH,), 8 the surface roughness, fs the Darcy swirl friction factor, and f) the

Farming friction factor. The reduction of the analysis parameters to F_'gun parameters takes the same form as that of"

Sparrow and Hail-Sheikh (1966):

(-dp)2( e ) Dp 4y,

fa = Cf = Pt--"_-Z ) 7 t __ g ) 0 5(--_o __ )2 - "_. Ergun

where dp --->Constant > 0 (A44)

dz

Ree = t.T) Tt.1---77)t.-_ -7 7 Ergun

(A45)

However, the data presented in figure 2 of Gambill and Bundy (1962) follow the parameters of Hong and Bergles

(1976):

1.09

Re = ReHong_Bergle s ---> -- XErgun ---> 1.12XErgun

g

(A28a)

fi = 7 = Cf, Hong-Bergles (A29a)

ft' = Cf,Hong_Bergle s = 2.314YErgun (A3 la)

Equation (A43) represents the best fit of a large data set from several investigations, yet at Re e = 10 000 and 8/I) =

6×10 4 the ratio of friction faclor data at y = 1.12 to those al y = 0.28 (an unusually tighl twist) is nearly 17. From

equation (A43) the ratio is 6. Although this difference has not been resolved, il is important to recognize that at a

given Reynolds number and value of y (noting that HID = y/D) the effect of surface roughness is very imporlant

because both increase the friction faclor.

Sparrow and Haji-Sheikh £1966).-------For the work of Sparrow and Haji-Sheikh (1966) on flow and heal transfer

in an arbitrarily shaped tube, the scaling factors lbr the half-robe configuration become

NAS A/TM-------2002-208914 16
,,H

CfRe _.---_-z )_. 0.-_p u 2 )_.---_-) = 63.06 (A46)

After substituting equations A2) and (A7) and recalling our assertion that @de-+ AP/L, equation (AI) becomes

(-@)2( e ) Dp 4y,

Cf = P_---_z )-_ 7_ g ) 0.5{oU.._2 =_ Ergun

where -@ --4 Constant > 0 (A44a)

dz

Re _. e )3 _.l-eA, _ ) -3 Ergun

YErgunXErgun = 70.94

Sparrow mid t taji-Sheikh (1966) 'also estimate entrance losses as

Cf + K - 0.5pU2as

where

(A45a)

(A39a)

(A47)

K= 2 1- dA

A

and for the half-tube configuration

K --> 1.463 (A48)

The agreement between Sparrow and ttaji-Sheikh ' 1966) and [}ate (1974) for the half-tube or single-twisted-tape

configuration is not surprising as they are within 1% of the values of Weigand (1948), which we discuss in the

section Torsion-laminm" flow analogy.

Bird et al. (1960L--For laminar flow in a tube (limit e--->l), Bird et al. (1960) give the average flow velocity as

avR2

u - (A49)

2gL

Substituting equations (A2) and (AT) and multiplying both sides by p2U o give

2 05[- p AP-g-_3 ]D2(G°'_( L°'_

Go = • L36(l_e)2] t-/7oJtT; (A50)

NASAffM--2002-208914 17
1 = 0.5 [L3---6/YErgunXErgun (A51)

YErgunXErgun = 72 -L where --_-L_ 1

Lo Lo

(A52)

which is in good agreement wilh the results of Date (1974), Sparrow and Hail-Sheikh (1966), Hong and Bergles

(1976), and Weigand (1948) for laminar flows in the halgtube or single-lwisted-tape configuration.

From data for flows in packed beds Bird et al. (1960) give 2L/L o = 25/6, resulling in the Blake-Kozeny equation. Kaviany (1995) gives 2L/L o = 5. From these results we anticipate that the bed correction factor for lairfinar

flows in a configuration with N twisted tapes in a robe will be largely independent of the twist for

= _ _._____K_X

"e a e < 150 where XErgun < 400 / 3 and e _ 1 (A53)

with some correction lk)r flow path length 2L/L o. However, the correction factor will not be as strong as that lk)r a

randomly packed bed, where XErgm_ < 10 is considered laminar.

At high Reynolds numbers the results of Smithberg and Landis (1964) show significant effects of twist Ho/D as

discussed earlier (eq. (A21)). At even higher Reynokls numbers the surface roughness will promote both eddy bursts

near the walls and separation effects. For commercially rough tubes (Schlichting, 1955) and in the lirnit e--_l (i.e.,

without a twisted tape in the tube)

(100)_ Re)Nikuradse (1933) = 400f{2a'_XErgun[..__) -5- 400 YE_unj2 XE_un= 1.27 (A54)

YErgun --->Constant --> 0.014 for e _ 1 (A55)

and 0.014 is the commercial rough-tube equivalent. From the data of Smithberg and Landis (1964) the trends lk)r the

single-twisted-tape or half-tube configuration, using D H, appem" to approach equation (A55).

Torsion-lm_finar flow analogy.--Weigand (1948) provides solutions to the problem of torsion in prismatic

members where the functionj(._,y) satisfies (see also fig. 2(g))

V2f = -1 (A56)

with fba r = 0 on the boundary of the cross section. This t_nction gives the torsion constant Jd for a member of cross

section A, where

Jd = 4 II fdA (A57)

The analogy between fully developed flow in a tube and torsional problems starts with the Navier-Stokes

equations

D____u= F - Vp / 9 + vV2u (A58)

Dt

NASA/TM---2002-208914 18
assumingsleadyflowwhereinertiaandbodyforcesaresmallandVp=Constant. Itthenfollowsthat

V2u=-1 (A59)

withZlbar =0Ol1theboundarywhere

ff u_

Ubax --

A

(A60)

and

u= dz 4g E/ /21 1- =C O 1-

(A61)

The equations describing this type of flow are in a form similar to those tbr torsion in a prismatic bar as

described by Weigand (1948) and pointed out by K.C. Cheng in a discussion to the work of SpmTow mad ttajiSheikh (1966). This leads to

R2 V2u =-1 = V2f

4G

(A62)

(A63)

/'/bar- _- -IIf = A 1

(A64)

Now the coefficient of friction mad Reynolds number are related as

Cf Re =

-dp De

( dZ I l(PUbaxDel_ _-g )(-dP']D2e

_/k ; ) 0"5Ubax"

J

(A65)

Substituting for Uba_ for full3; developed lmalinar flows, where DH = D e, !gives

Cf Re - 8AD2

Jd

(A66)

Subslituting for

NASAfI'M--2002-208914 19
De m

4A

Perimeter

(A67)

gives

128A 3

CU Re = Jd (Perimeter) 2

(A68)

where

A=_R2Ii-(2°_-sin2°_]]k _ )J (A69)

Perimeter = 2gRI 1 o_-sin o_ ]

(A70)

For circular tubes (g = _ ---(cz = 0) --> _) Weigand (1948) gives in his table IV

= J---@= 1.571 = re

R 4 2

(A71)

and

CfRe = 64 (A72)

For semicircular tubes [Z = rc ---(ct = re/2) --->rc/2] Weigand (1948) gives in his table IV

(A7 la)

R 4

Cf Re = 16_ Jd - 62.97 (A73)

CfRe] = 63.06

"Sparrow Haji-Sheikh (1966)

(A46a)

Although the (.)Re values of Sparrow and Haji-Sheikh (1966) are monotone with cz, those of Weigand (1948) are

not. However, all are well represented by

128A 3

Cf Re = 63 = Jc/(Perimeter) 2 (A74)

and would be a ,good engineering_ approximation for laminar flows in tubes of vm'ious cross sections _(e,._,.,°semicircular). For Date's work, using the preceding relations,

NAS A/TM-------2002-208914 20
C,Re=4_(_-)Re,(_)=4_Re,(_)2=63 (A75)

and for the twisted tape under the fbllowing conditions:

H

y =-- ---> oo;

D

D 2

cz=--" --=1+--

2 D e

Re/= 42.19 (A76)

Generalization.--For tubes with single twisted tapes

(2e -]l-n

XErgun YErgun =70"9+[gl(-_)l_"_-XErgun )

(A77)

where from Smithberg and Landis (1964), tbr XEsgu, > 2600/3,

b(-_/]=oo4_+_,(_-o,/-'2 (A78)

and

1

and ['or 400 < 3XErgun < 2600

Ol 1+2 X

oo+ +o (A78a)

n= 1

Some consolidation in the turbulent flow regime follows from Smithberg and Landis (1964):

6.9

f -0.9+--

fH/D--+_ Ho

D

(A79)

where

0.046

ffl/D_ - ReO. 2

(A79a)

NASA/I'M--2002-208914 21
andisthoughtofasthenonideal flowpathlengthL for a particle to travel a bed or robe length of (Lo)II/D.____. Thus,

gl(I-111)) can be modified as

g = 0.9-_ 2H

D

(A80)

Lopina and Bergles (1969) _)f equal interest is the simple relation given by Lopina and Bergles (1969) for

turbulent flows in smooth tubes C+__20%' *or water and air for 5x103 < Re < 4.5×105 and _ < H/D < _I_:

f= 2.75(H) -0"406 (A81) fo

where,, is the isothermal friction factor for an empty lube,f_ = 0.046/Re 0%_' Reducing the results of Lol?ina and

Bergles (1969), with equation (A9), gives

(zSdPDiP_(GDi_(DF g 1--El2 E

fRe=k2LG2)t _ )(Dpel-e)

(A8 la)

where

fo Re/f/= (H_-°'4°6(1+2) 2

_fo) XErgunYErgunO'61_. --'_) e 3

(A81b)

(A8 lc)

Re = 2(1 + 2)XErgun (A81d)

The resulting fit to lhe dala is lower than the data of Smilhberg and Landis (1964) by about 10% for HID = 2.48. So

the agreement can be good and the simplicity may be of merit for a limited Reynolds number range.

We also note that for large values of XErgun the data will become independent of XErgun as equivalent surface

roughness (k s ,-: 8/1)) plays a role. Thus, for large Xl_;r_.un fbr a given geometry

YErgun = g2 -_, s = Constant (A82)

or in generalized terms

XErgunYErgun =70.9+Ig2(H,ks)lXErgun (A83)

NAS A/TM-------2002-208914 22
andfromextrapolating thedata

XErgungErgun = 70.9 + 0.014XErgun (A83a)

Porous Media Flows and N Twisted Tapes

For porous-media flows Ergun (1952) combined the turbulent and laminar data, noting that at high Reynolds

numbers the friction factor approached a constant as is consistent with flow in a fully roughened tube (k s > 0). The

result was a linea" sum of the Kozeny a_d Burke-Plummer equations, and as illustrated in Bird et al. (1960) the

generalized form may be written

XErgunYErgun = 150 + 1.75XErgun (A84)

And if the Ergun data were for a single insert, and following the development on page 199 of Bird et al. (1960),

Dv 83 1 OVoDv

XErgunYErgun = 72

which nearly coincides with the 70.9 of Lopina and Bergles (1969) in equation (A83), indicating that friction factors

for flows in tubes with multiple twisted tapes should be much higher thaa_ measurements are showing.

For tubes with N twisted tapes the laminar data are below, yet parallel to, the XErgunYl__;rgunrelation for flows in

porous media. From the data in table 1 (which may yet require Prandtl number or viscosity corrections),

XErgunYErgun = Constant =' 45 (A85)

with a suggested general form to_ N tv_lsted tapes in a tube following equation (A83), which appears as

XErgunYErgun = 45 + 0.009XErgun (A85a)

From equations (A93) and (A95) for parallel flows in cylindrical fibrous materials, with e = 0.61 for a 48-twistedtape cylin(kical bundle,

XErgunYErgun = 36k = 46.6 (A85b)

suggesting that pressure losses in the twisted-tape bundles are less than in other porous-media flows.

The combination of implied losses and those of, for example, cotton fibers, suggests a flow vortex structure that

impedes the passage of fluids more than hair or glass fibers do. This structure does not suggest a direct relation to

either heat or mass transfer. These implications remain to be investigated.

Fibrous Bulk Materials

The work of Fowler and Hertel £!940) for flows through wads (e.g., wool, glass wool, cotton, rayon, kapok)

provides a solution:

GO _.-_)_.-_) _77-7_)_----_x ) (as6)

NASA/TM--2002-208914 23
where0.18<k < 0.2, ToP = p, and the specific surface area a_ = "c/_ equals the ratio of the element volume "c to the

element surface ¢5. Equation (A86) can be rearranged to the Ergun form:

1 k(_-]2( E3 /( D /(_xP/ [ (_! 1YErgun _ _ (1-e)bt = k z 2

XErgun -G-7o_p/) ';)[_-g)[--G--fo)_--"_t)J[Dp J

(A87)

where

and

(A16)

Dp= 6 =6_c (A88)

av (5

For k = 0.2

YErgunXErgun = 180 (A89)

as also recommended by Kaviany (1995). For k = 0.18 as recommended by Fowler and Herlel (1940)

YErgunXErgun = 200 (A90)

The work of Sullivan (1942) for parallel fibers illustrates a distinct departure from the relation YErgunX_gun =

Constant. Sullivan's relations are similar to those of Fowler and ttertel (1940) and follow the same reduction to the

Ergun form:

6 6

- - (A91) DV So

XErgun

(A92)

YErgun XErgun = 36k (A93)

where the units of Sullivan (1942) are in the cgs system (dyne.. _,,,,cm, sl.

k= k°

g

(A94)

and 9 ---(sln'-_)avg, where _ is the angle between the interface normal and the microscopic flow. For flows with

p_'allel "cylindrical" fibers and _--> 1

NAS A/TM-------2002-208914 24
g

kg --> k o - 3

1.02 - e

4- 0.8 (A95)

For flows with parallel cotton fibers and a < 0.85

kg --> k o = 2.5 (A96)

For e > 0.85, k(-->k o tends to follow ko,paralM.

The ggrounXgroun product R)r parallel flows in cotton fibers with e < 0.85 is about half that cited by Fowler and

Herlel (1)4()_ ( for packed #o wads (90 versus 180 or 200 depending on the value of k used for packed wads of fibers)

and would agree with Fowler and Herlel for packed-wad flows where e --+ 0.95. The k used by Sullivan is the

inverse of that used by Fowler and Hertel.

Hersh and Walker (I980) revisited the work of Sullivan (1942) and provided a correlation over the range of data

as follows, where the units are in the cgs system (dyne, g, cm, s):

For flow parallel to fibers (0.1 < _ < 0.985)

2fRe= Apd2 = 15"74(17 e)1"413. gLgo [1 + 27(1- g)3] = fpaxallel (g)

(A97a)

For flow perpendicular to fibers ((I.7 < _ < 0.992)

APd 2 [.(1- ,3/2 3

2f Re - _ - 64 [ ee-', ill + 14.75(1- e)3 l = rperpendicula s (g)

(A97b)

where the laUer expression was developed by Davies (1952).

After refitting the data of Sullivan (1942), lhe Hersh and Walker (1980) form may be re-expressed as

(Apd2 I = 3___+08/16(1-g]2=2(Apd2 I

_, )perpendicular

_,gLU°)parallel 1.02-e " J e \ e ) 3 gLU o

(A98)

And the Ergun lbrm follows:

1

(YErgunXErgun)parallel = 36k=-3 (YErgunXErgun)perpendiculax (A99)

Although these expressions generally agree with those presented by Hersh and Walker '1980), an average of the two

forms more closely approximates Sullivan's (1941) data (see figs. 2 and 3 in ttersh and Walker, 1980). These

modified tbrms (eqs. (A97) and (A98)) are presented herein as figures 11 and 12.

Note that tier parallel flows through cotton fibers the shape factor is 2.5, and for flows though fibers aligned

normal to the flow (Davies, 1952) it is 3. The implication is that flows though fibrous materials (e.g., cotton) Nighed

with the flow do not differ significantly from flows through fibrous materials aligned normal to the flow. ttowever,

flows through fibrous materials still have three times the pressure drop as flows along parallel cylindu'ical materials

when other parameters remain fixed. Sullivan (1941) tbund that the pressure drop through packed beds with fibers

oriented perpendicular to the flow was double that of flows through parallel fibers.

NASA/TM--2002-208914 25
ReducingtileHersh-Walker (1980)formtotheErgun(1952)form, notingthatSullivangives(d)asfollows

fromequations(A6)and(A7),resultsin

where

Substituting, equation(A97)becomes

4 6(1-e) (AIO0)

(d) 4 - 2Dp (AIO1)

& 3

( 2Dp _ 2

(A102)

XErgunYErgun = F(e e (A103)

For flow parallel to the fibers (0.1 < _ < 0.985):

YErgunXErgun = 35.415{[(1-e)-°587e2][1+27(1-e)3]} (1104)

and, for reference, as _ .9 0.27 the quanlity in braces approaches 1. Equalion (All 04) can be represented by

0.75 < e< I.

(XErgunYErgun)1/4 = 1.6 - 0.7 log e (1 - _) (A 104a)

For flow perpendicular to the fibers (0.6 < e < 0.992):

YErgunXErgun = 144{[(1-e)-0"5]e2[1 + 14.75(1- e)3]} (A105)

Equation (A105) can be represented by 0.75 < _ < 1.

\1/4

XErgunYErgun ) = 2.5- 0.81Oge(1- g ) (A105a)

and for this case the quantity in braces approaching 1 as e -9 0.5 is beyond the region of validity of the relation.

From figures 11 mad 12 (replotted figs. 2 and 3 from ttersh and Walker, 1980), the relations appem" to be in

good agreement, yet the sensitivity to small changes in e becomes paramount. Compared with the twisted-tape data

(see table 1) for e = 0.522 and N = 48 twisted tapes in a cylindrical (pm'allel) bundle, equation (A104) gives

XErgun}_rgun = 58.8, which is nearly 7/3 greater than the data (see eq. (A85)). As the bed porosity approaches that of

a filter or a particulate sepaa'ator, the XErgunYErgun product is not a constaaat but depends on the filter porosity, as

noted by ttersh and Walker (1980) and Sullivan (1942).

NAS A/TM-------2002-208914 26
Silverman and First (1952) reported data for edge filtration of 0.5-btm dust and a baby-oil smoke generator with

average 0.6-btm particulates. The filtration materials were a variety of felts, flocked papers, and fiberglass media.

They noted that rockwool, silica fiber, or aluminum oxide fiber could be used at high temperatures. For their

correlation

YErgunXErgun = 9 29e3 (1- e) 1"4 (A106)

Davies (1952) provides a theoretic_ foundation for several types of pm'tictflate separator, such as settling chambers,

conical elutriators, inertial and cyclone separators, .jet impingement mechanisms, precipitators, thermal separation

mechanisms, and filters. Data for flow through fibrous materials fbllow:

YErgunXErgun = 9(70)E(1- g)l'5[1 + 52(1- g)l.5] (A107)

The Davies 1952) results range up to three times higher than those of Silverman and First (1952) to several times

higher than those of ttersh and Walker (1980).

Using the data in table I for 48 twisted tapes in a cylindrical parallel) bundle, where a = (I.522, gives,

XErgunYErgun = 58.8 from eq. (A104) (A108a)

XErgunYErgun = 46.6 from eq. (A85b) (A108b)

For the 48-twisted-tape data prediction

46.6 + 58.8

____XFrgunYFrgun-- -- 52.7 (1109)

2

or less than 1.2 times that of the reported data (table I and eqs. A85)).

Packed Beds of Spheres

Wentz and Thodos (1963) measured the pressure drop across packed (cubic, body centered, and face centered)

and distended bands of five layers of 3.12-cm- (1.23-in.-) diameter spheres held in place by short wires in drilled

holes and epoxy. Spheres in the distended models were separated to simulate bed swelling. Both sets of data were

correlaIed by

0.396

YErgun - v0.05

aErgun - 1.2

2550 < XErgun < 64 900 (All0)

which is less than half that of the Ergun equation (A84 but over a lm'ger range of XErgun. As the bed length was

short (five spheres), XErgun dependence may be attributed to as-yet-undeveloped turbulent flow. A problem combining or extending this relation occurs with the laminar regime because a singularity occurs at XErgun = 38.34.

However,

0 84

XErgunYErgun = 150 + 4.5Xt_rgun (A111)

is a fbrm that includes this high-XErgun turbulent regime, yet is higher than the Ergun equation (A84 fbr the

transition region 15 < XErgun < 300 and as much as 14% higher fbr the region 40 < XErgun < 80 (fig. 13). The

NASAfI'M--2002-208914 27
turbulent-flowpressuredropsthroughthesepackedbedsofsphereshavesimilarXFrgun dependence yet are 30 to 40

times larger than R)r a single twisted tape in a tube over similar ranges in XFreu n. The expression

XErgunYErgun = 0.031150 + 4.5XOr84n] (All2)

provides a reasonable fit to the data of Smithberg and Landis (1964) and Koch (1958) in figure 14.

Some Sample Calculations

The basic parameters for 48 twisted tapes in a 2.54-cm- (1.0-in.-) diameter tube are as follows:

Fluid:

p = 0.9 g/crn 3

bt = 0.9 g/cm-s

Tapes: {w} and {t} represent average values of twisted-tape width and twisted-lape thickness measured at each end of

tile 48 twisted tapes (table 2).

(w} = w = 0.3234 cm (0.1273 in.)

(t) = t = 0.1275 cm (0.0502, in.)

D o = (w 2 + t2) °'5 = 0.348 cm (0.137 in.)

Three full 360 ° twists in 16.5-cm (6.5-in). length provides an average twist ratio of

16.5

H___q_o = 3 = 15.8

DO 0.348

H

--=7.9

Do

L o = 16.5 cm

Other parameters: @exp} = 0,61 is the average area-weighted porosity (0.593 < Cexp < 0.628).

8exp = 0.61

3 D(1 - e) (1.5)(2.54)(0.39)

Dp- l+22N(t+W)_D 1+(2)(48) (0"1275(_-. 54_- + 0"3234] ) - 0"231 cm

NAS A/TM-------2002-208914 28
dV dV 0.9

P

Go_ dt _ dt 60 _2.96x10-31)- wherel),isincm 3

A o 5 •067

min

AP = dP(6.8947 x 104) in g / cm - s 2, where P is in psla

XErgun GoDp I 0"231 ]=0.659Go - (1---_g - G° "(0•39)(0•9)

where Go is in g / cm 2 - s

)_(0.9)(0.61)3 0.21A 

rE'gun=t.Oo2 Jt )tl- J 0.39 16.52

- 73.34 x 10 .4 dP

150

Ycalc - + 1.75

XErgun

Data

Date Geometry dV/dt,

cm3/min

12-10-98 Borda + 5900

screen

12-14-98 Orifice + 4754

screen

12-14-98 Orifice - 6767

screen

12-14-98 Borda + 6777

screen

12-19-98 Orifice + 6556

screen

AP, Go, de, XBrgu_ _'Brgu_ Ycalc YcJ

psi g/cm2-s g/cm-s2 YErgml

2.54 17.46 17.5x104 11.5 4.22 14.8 3.51

1.88 14.07 13.0x104 9.27 4.81 17.9 3.73

2.47 20.0 17.0x104 13.2 3.12 13.1 4.21

2.86 20.0 19.7x104 13.22 3.6 13.1 3.64

2.48 19.4 17.1x104 12.79 3.33 13.5 4.04

For all the data (see table 1)

Ycalc - 3 6 +0.7

• -0.6

YErgun

Standard deviation, 0.32

YErgunXErgun = 45 -5+6 Standard deviation, 3.5

For the minimum and maximum twisted-tape thickness and wkith, from table 2, and the combined data set of

lable 1,

(t) (w)

0.1235 cm (0•0486 in.) 0•318 cm (0.1252 in.)

0•1296 cm (0•051 in.) 0•3287 cm (0•1294 in.)

@exp)

0.6276

0.5934

Ycalc/YErgun Standard

deviation

3.2 0.28

4 0.35

NASAfI'M--2002-208914 29
where(Eexp} isthearea-averaged porosityand

39<YErgunXErgun <51

However, thedatasetsforlheBordandorificewithscreenandtheorificewithoutscreenhavedislinctflow

characleristics, andmorerepresentalive valuesoft Frgun,YFrgun can be determined. For the 48-twisted-tape data of

table 1, the coefficients of the least-squares trend line

lOgl0[gblrgun] = A 1 1Ogl0[XErgun ] + B l

through individual data sets are tabulated below, where R 2 is the regression coefficent.

Flow configuration A1 B 1 R2 Constant _ YErgml XErgml

Average porosity = 0.61

Bordawith screen _ -0.9873 1.6673 0.9965 46.5

Orifice with screen -1.0006 1.6436 .9984 44.0

Orifice without screen -0.9982 1.6132 .9988 41.0

Combined data sets -0.9867 1.6469 .992 44.4

Minimum porosity = 0.5934

Bordawith screen a -0.9879 1.6202 0.9964 41.7

Orifice with screen -1.0006 1.5958 .9984 39.4

Orifice without screen -0.9982 1.5655 .9988 36.8

Combined data sets -0.9869 1.5994 .992 39.8

Maximum porosity = 0.6276

Bordawith screen a -0.9866 1.7190 0.9966 52.4

Orifice with screen -1.0006 1.6961 .9984 49.7

Orifice without screen -0.9982 1.6658 .9988 46.3

Combined data sets -0.9864 1.699 .9919 50.0

aone questionable data point set at average of previous and following points in table 1.

Estimates from figure 6 of Smithberg and Landis (1964) lbr air and water data are

3 Re

XErgun= 2

YErgun = 3f

NAS A/TM------2002-208914 30
Reynolds number, Re

5000 10 000 20 000 40 000 60 000

XErgml

7500 15 000 30 000 60 000 90 000

fat Ho/D = 3.62 29x10 3 21x10 3 15.5x10 3 13x10 3 12.5x10 3

YErgml 87X10 3 63X10 3 46.5X10 3 39X10 3 37.5X10 3

fat Ho/D = 4.34 22.5x10 3 16.5x10 3 13x10 3 11xl0 3 9.7x10 3

YErgml 67.5X10 3 49.5X10 3 39X10 3 33X10 3 29.1X10 3

fatHo/D= 10.3 14x10 3 llxl0 3 8.5x10 3 7.4x10 3 6.8x10 3

YErgml 42X10 3 33X10 3 25.5X10 3 22.2X10 3 20.4X10 3

fat Ho/D = 22 12;<10 3 9.9x10 3 7.6x10 3 6.6;<10 3 6;<10 3

YErgml 36X10 3 29.7X10 3 22.8X10 3 19.8X10 3 18X10 3

fat Ho/D = _ 8;<10 3 7.1;<10 3 6;<10 3 5.35;<10 3 5;<10 3

YErgml 24X10 3 21.3X10 3 18X10 3 16.1X10 3 15X10 3

The Smithberg and I,andis (1964) data (N = 1) are Reynolds number dependent. It would be interesting to determine

if data for N = 2, 3 ..... 48 form parametric families that become more independent of Reynolds number as N increases (see eq. (A55)).

Estimales 1YoreKoch (1958) as provided l'rom figure 7 of Sn-tithberg and Landis (1964) are as follows:

Reynolds number, Re

2000 3000 6000 10 000 20 000 30 000 50 000

XErgml

3000 4500 9000 15 000 30 000 45 000 75 000

fatH/D= 5 32.5;<10 3 26.5X10 3 18;<10 3 14.6X10 3 ll.6x10 3 10;<10 3 9;<10 3

YErgma 97.5X10 3 79.5X10 3 54X10 3 43.8X10 3 34.8X10 3 30X10 3 27X104

fat H/D = 8.5 21x10 3 17.5x10 3 13.2x10 3 10.9x10 3 8.6x104 7.5x10 3 6.4x10 3

YErgma 63X10 3 52.5X10 3 39.6X10 3 32.7X10 3 25.8X10 3 22.5X10 3 19.2X10 3

fat H/D = 22 14.2x10 3 12.3x10 3 9.7x10 3 8.5;<10 3 7.1x104 6.4;<10 3 5.7;<10 3

YErgma 42.6X10 3 36.9X10 3 29.1X10 3 25.5X10 3 21.3X10 3 19.2X10 3 17.1X10 3

fat H/D = _ 10;<10 3 9.5;<10 3 8.1;<10 3 7.4;<10 3 6.5x104 5.9;<10 3 5.4;<10 3

YErgma 30X10 3 28.5X10 3 24.3X10 3 22.2X10 3 19.5X10 3 17.7X10 3 16.2X10 3

NASAffM--2002-208914 31
_0

II

©

O

O

O

t"q

A

A

O

?

O

?

t",l _'h

Gh Gh

',D

G'5

H

H

_9

p_

G_

©

Gh G_Gh_._:__ _' t..'_-_"__GhG_ _'

<__< _< _ _<< <

II II II

o.. _ _:_1 _::::_

_ _1_ _ =_, _ _+

_"_1 "_ II c_ll_ II II

p_

G_

¢D

_D

G_

_9

_9

G_ "_"

eq

+

o

,- +

H

II "

#

O _D G_

NAS A/TM------2002- 208914 32
0 0 0 0 e_h e_h e_h

< < < < < <<<

0

0 _C_

cq c_

II II

m

_ _ _1 _

II

+

I

II II

?

t_

?

t¢'5

+ _ I

U'3

II _ ,_ _,o

II II II II II I

_ _ j ___" _ "

_ _ _ _ _

c_

c_

A' A'

c_ c_ c_

_> _> _>

i

_D

C_

II _n

NASA/I'M--2002-208914 33

A

A 1

A o

A s

a

a v

B 1

q

Co

D

D e

D H

D i

D O

z)p

Dsphere

Dmbe

d

eiso

F(C)

f

L

L

L

G

Go

g

go

gl

g2

H

14o

Jd

K

k

ko

APPENDlX B

SYMBOLS

flOW area

constant

tube cross-sectional area without twisted tape

surface area or area of sphere

ratio of wetted surface area to bed volume

specific surface area, z/c_

constant

flow coefficient of friction

constant relating velocity profiles to radial position

tape diameter or width (for thin tubes, same as flow tube diameter)

equivalenl diameter

hydraulic diameter; characteristic length of packed bed

inside diameter

virtual dia_rmter of twisted tape

equivalen{ particle diameter

sphere dia_rmter

flow lube diameter

fiber dia,_mter

average fiber diamemr

subscript denoting equivalent isothermal, eq. (A43)

function relating friction to flow direction, eqs. (A97)

Farming 1}iction factor, [,_)_ 0.-_pu 2 )

no-swirl friction factor, eq. (A43b)

friction factor, eq. (A75)

Fanning friction Nctor for tube without twisted tape

Darcy swirl (twisted tape) friction factor, f_. = 4f (Darcy f= 4 Fanning J)

mass flow, pu

mass flux, pU o = W/A o

function defined in eq. (A24)

function defined in eq. (A23)

function defined in eqs. (A78)

function defined in eq. (A82)

tape twisl through 180 ° or one-half full wave

tape twist through 360 ° or one full wave

torsion constant

entrance pressure loss parameter, eq. (A47)

packing constant, ko/_, eqs. (A86) and (A94)

porosity shape parameter, eq. (A94)

NASA/TM--2002-208914 35
k S

L

Lo

M

m

N

n

n t

P

P

R

R h

Re

Re e

Re i

F

G

S

S o

t

(0

U

G

G

Z_

Z_ba r

V

Vs

Vsolid

Vtotal

?2

?20

v t

W

W

(w)

XErgun

G

X

Ycalc

surface roughness parameter

equivalent flow or twisted-tape length; nonideal flow path length

tape length without twist; straight-line distance between pressure taps; bed or tube length

number much greater than unity

exponent, eq. (A42)

number much less than unity

number of twisted tapes in tube

exponent, eqs. (A21) ea_d(A43)

number of 271;twists

pressure, experimental

pressure, calculated

outer radius

ratio of bed cross section available for flow to wetted perimeter

Reynolds number, puD/u

equivalent Reynolds number

Reynolds number, eq. (A75)

radial position

outer "wall" radius

wetted perimemr; surface area of fibrous specimen

surface area, eq. (A101)

tape thickness

average twisted-tape thickness

bulk average axial velocity

equivalent fluid velocity

empty or unpacked-bed velocity

velocity, general or vector; superficial velocity

average velocity

bed volume

volumetric flow rate

volume of sphere

volume of solid in porous bed

total bed volume (solid plus void)

flow velocity in fibrous specimen

circumferential velocity; tangential fluid velocity

velocity at ro

tangential velocity

mass flow rate

twisted-tape width

average twisted-tape width

Ergun Reynolds number parameter, GoDp/(1 - e)_t

pressure drop data parameter

axial position

calculated Ergun parameter

NAS A/TM-------2002-208914 36
Yt;rgun

Y

Z

Zo

z

8

Yo

E.

Cexp

E;model

I¢

P

gt

v

(p

(5

Ergunfrictionf'actorp_ameter,(pAP/_o_)[_ _/(1-_)](D_/Lo)

flow data parameter

number of tube diameters per 180 ° of twist (also used as coordinate, fig. 2(e))

analytical axial locus

equivalent axial distance without twist

axial coordinate

polar coordinate parameters, fig. 2(e)

surface roughness

bulk density parameter, eq. (A86)

bed porosity parameter

experimental bed porosity

modeled bed porosity

torsion parameter, eq_ (A71)

average fluid density

viscosity

g

-- kinematic velocity P

analytical twist parmneter

angle between interface normal and microscopic flow

surface-to-volume pm'ameter, eq. (A86)

thickness of fibrous specimen

deviation

porosity parameter, eq. (A94)

NASA/I'M--2002-208914 37

REFERENCES

Bergles, A.E. (1998) Augmentation of Heat Transl\:r, 2.5.11 in Heat Exchanger Design Handbook 1998, G.F. Hewitt, ex. ed.,

Part 2 Fluid Mechanics and Heat Transfer, Begell House, Inc., New York.

Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960) Transport Phenomena, John Wiley 8::Son, New York.

Braun, M.J., Batur, (2., and Karavelakis, G. (1988) Digital hnage Processing for Quantification Through Full Flow Field

TracMng (FFFT) in Narrow Geometries at Low Reynolds Numbers, AIAA Paper 88.---3781.

Date, A.W. (1974) Prediction of Fully Developed Flow in a Tube Containing a Twisted Tape, Int. J. Heat and Mass Trans., vol.

17, pp. 845-859.

Davies, C.N. (1952)The Separation of Airbourne l)ust and Particles, Proc. Inst. Mech. Engrs. (London), t31, pp. 185-198.

Ergun, S. (1952) Fluid Flow Through Packed Columns, Chem. Eng. Prog., vol. 48, no. 2, pp. 89-93.

Fowler, J.L. and Hertel, K.L. (1940) Flow of a Gas Through Porous Media, J. Appl. Phys., vol. 11, pp. 4.96-502.

Gambill, W .R. and Bundy, R.D. (1963) High-Flux Heat Transfer Characteristics of Pure Ethylene Glycol in Axial and Swirl

Flow, A.I.Ch.E. Journal, vol. 9, no. 1, p. 55.

Gambill, W.R. and Bundy, R.D. (1962) An Evaluation of the Present Status of Swirl-Flow Heat Transfer, ASME

Paper 62-t lT-42.

Gambill, W .R., B undy, R.D., and Wansborough, R.W_ (1961) Heat transl\:r, Burnout, and Pressure Drop for Water in Swirl Flow

Through Tubes With Internal Twisted Tapes, Chemical Engineering Progress, Symposium Series No. 32, vol. 57, pp. 127-137.

(Also ORNL---2911, Apr. 1960.)

Gambill, W.R. and Greene, N.D. (1958) Boiling Burnout With Water in Vortex t!qow, Chem. Eng. Prog., vol. 54, no. 10,

pp_ 68---76_

Gilinour, C.H. (1958) Nucleate 13oiling--A Correlation, Chem. Eng. Prog., vol. 54, no. 10, pp. 77-79.

Hersh, A.S. and Walker, B. (1980) Acoustic Behavior of Fibrous Bulk Materials, NASA Contract NAS3-21975(A80-35951),

AIAA Paper 80---8096.

Hendricks, R.C., I,attime, S., Braun, M.J. and Athavale, M.M. (1997) Experimental Visualization of Flows in Packed Beds of

Spheres, Proceedings of the First International Symposium on Flow Visualization and Image Processing. (Also NASA

TM-107365.)

Hong, S .W. and Bergles, A.E. (1976) Augmentation of Laminar Flow Heat Transl\:r by Means of Twisted--Tape Inserts, J. Heat

Trans., vol. 98, no. 2, pp. 251-256.

Kaviany, M. (1999) Principles of Heat Transl\:r in Porous Media, Second Ed., Springer--Verlag, New York.

Koch, R_ (1958) Druckverlust und Wfirme{ibergang bei verwirbelter StriSmung. VDI--Forschungshaft 469, series B, vol. 24.

(Referenced in Smithberg and Landis, 1964).

Lopina, R.F. and Bergles, A.E. (1969) Heat Transfer and Pressure Drop in Tape-Generated Swirl Flow of Single-Phase Water,

J. Heat Trans., vol. 91, Aug., pp. 434-442.

Nikuradse, J. (1933) StrOmungsgesetze in rauhen Rohren, Forschg.--Arb. Ing.--Wessen, no. 361. (Rel\:renced in Schlichting,

1955).

Schlichting, H. (1955) Boundau Layer Theory, McGraw-Hill, New York.

Silverman, L. and First, M.W. (1952) l!Mge and Variable Compression Filters for Aerosols, Industrial Engineering and Chemistry, vol. 44, no. 12, pp. 2777-2783.

NASA/TM--2002-208914 39
Smithberg, E.andLandis, [7.(1964)FrictionandForcedConvection Heat-Transfer Characte,_stics inTubesWithTwistedTape

SwirlGenerators, J.tteatTrans., vol.86,no.1,pp.39-49.

Sparrow, E.M.andHaji-Sheikh_ A.(1966)FlowandtteatTransfer inDuctsofArbitraryShapeWithArbitrau¢Thermal

Boundary Conditions, J.HeatTrans., vol.88,seriesC,pp.351---358.

SullivamR.R.(1942)SpecificSurfaceMeasurements onCompact BundlesofParallel Fibers, J.Appl.Phys., vol.13,Nov.,

pp.725---730.

SullivamR.R.(1941)Further StudyoftheFlowofAirThroughPorousMedia, J.Appl.Phys., vol.12,June, pp.503-508.

Weigand, A.(1948)TheProblemofTorsioninPrismatic Members ofCircular Segmental CrossSection, NACATechnical

Memorandum 1182.

Wentz, C.A.andThodos, G.(1963)Pressure DropsintheFlowofGasesThroughPackedandDistended BedsofSpherical

Particles, A.I.Ch.E. Journal, vol.9,no.1,pp.81-84.

NASA/TM-------2002-208914 40
TABLE 1. FLOW AND PRESSURE DROP DATA AND ANALYSIS FOR 48 TWISTED TAPES IN CYLINDRICAL TUBE

fz, AlP, Go, dP, XErgun z

cm3/min psi g/cm 2 s g/cm s2 0.659G °

a8218 3.55 24.33 244762 15.28

7657 3.29 22.66 226836 14.23

5900 2.54 17.46 175125 10.97

4730 2.07 14.00 142720 8.79

3881 1.67 11.49 115141 7.21

2804 1.11 8.30 76531 5.21

2790 1.14 8.26 78600 5.19

1546 0.59 4.58 40679 2.87

1272 0.47 3.77 32405 2.36

b957 0.31 2.83 21374 1.78

975 0.37 2.89 25510 1.81

1518 0.63 4.49 43437 2.82

2014 0.87 5.96 59984 3.74

2713 1.13 8.03 77910 5.04

3538 1.59 10.47 109626 6.58

4307 1.94 12.75 133757 8.01

5260 2.33 15.57 160647 9.78

6129 2.67 18.14 184088 11.39

7151 3.09 21.17 213046 13.29

8207 3.49 24.29 240625 15.26

6354 2.73 18.81 188225 11.81

4435 1.92 13.13 132378 8.24

2471 1.02 7.31 70326 4.59

c1592 0.686 4.71 47298 2.96

2721 1.174 8.05 80944 5.06

4349 1.866 12.87 128655 8.08

4311 1.84 12.76 126862 8.01

6112 2.58 18.09 177883 11.36

8045 3.37 23.81 232351 14.95

7193 2.96 21.29 204083 13.37

6057 2.49 17.93 171678 11.26

4993 2.04 14.78 140652 9.28

3459 1.41 10.24 97215 6.43

2298 1.04 6.80 71705 4.27

1526 0.64 4.52 44126 2.84

958 0.39 2.84 26889 1.78

446 0.2 1.32 13789 0.83

d5859 2.52 17.34 173746 10.89

8156 3.48 24.14 239936 15.16

6777 2.86 20.06 197188 12.60

aBorda + screen (12/10/98).

bQuestionable.

°Borda + screen (12/12/98).

dBorda + screen (12/14/98).

YEr_n z

73.34x10 _

dP/Go 2

1.81

1.94

2.52

3.19

3.82

4.87

5.05

8.51

10.02

11.67

13.42

9.43

7.40

5.29

4.38

3.61

2.90

2.45

2.08

1.79

2.33

3.37

5.76

9.33

5.47

3.40

3.41

2.38

1.80

1.97

2.34

2.82

4.06

6.79

9.48

14.65

34.67

2.53

1.80

2.15

Ycalc =

1501/..7E_un+

11.57

12.29

15.43

18.81

22.54

30.53

30.67

53.95

65.19

86.07

84.51

54.91

41.82

31.49

24.56

20.49

17.09

14.92

13.03

11.58

14.45

19.94

34.41

52.44

31.41

20.30

20.47

14.95

11.78

12.97

15.07

17.91

25.08

36.86

54.63

85.98

182.68

15.52

11.64

13.66

YcalJYErgun (_1 XErgmlYErg un (_2

6.38 0.0039 27.69 3.42

6.35 0.0009 27.54 2.89

6.13 0.0356 27.60 3.08

5.90 0.1799 28.05 4.89

5.90 0.1797 27.58 3.03

6.27 0.0024 25.37 0.22

6.07 0.0606 26.19 0.12

6.34 0.0003 24.46 1.90

6.51 0.0353 23.68 4.65

7.37 1.1111 20.76 25.77

6.30 0.0005 24.32 2.30

5.82 0.2458 26.60 0.58

5.65 0.4437 27.69 3.42

5.95 0.1377 26.70 0.74

5.61 0.5088 28.81 8.80

5.68 0.4088 28.87 9.19

5.89 0.1888 28.39 6.52

6.09 0.0548 27.92 4.34

6.26 0.0041 27.70 3.45

6.48 0.0264 27.26 2.01

6.20 0.0151 27.54 2.89

5.93 0.1557 27.75 3.65

5.97 0.1205 26.46 0.38

5.62 0.4925 27.62 3.17

5.74 0.3320 27.66 3.30

5.97 0.1236 27.50 2.76

6.00 0.1054 27.36 2.31

6.28 0.0017 27.06 1.48

6.56 0.0581 26.85 1.02

6.57 0.0644 26.38 0.29

6.44 0.0144 26.35 0.26

6.35 0.0008 26.19 0.12

6.17 0.0221 26.13 0.08

5.43 0.7948 29.01 10.04

5.76 0.3087 26.88 1.09

5.87 0.2045 26.09 0.06

5.27 1.1047 28.74 8.43

6.13 0.0353 27.57 2.99

6.45 0.0181 27.35 2.28

6.36 0.0016 27.05 1.47

eOrifice + screen (12/14/98; reverse of Borda + screen).

_Orifice without screen (12/14/98; reverse of Borda without screen).

gOrifice without screen (12/19/98; reverse of Borda without screen).

hN 1 points.

iN points.

JThrow out one point.

NASAfI'M--2002-208914 41
V, AP, Go, dP,

cm3/min psi g/cm 2 s g/cm s 2

TABLE 1. CONCLUDED.

xBrg,_,, = Y_+,_,, = Y++,I° = Y++/gBrg,_,,

0.659G o 73.34×104 150/XErgun+

dP/Go 2 1.75

4546 1.9 13.46 130999 8.45

2367 1 7.01 68947 4.40

1768 0.75 5.23 51710 3.29

1245 0.55 3.69 37921 2.31

833 0.36 2.47 24821 1.55

718 0.32 2.13 22063 1.33

e982 0.35 2.91 24131 1.83

2239 0.88 6.63 60673 4.16

4754 1.88 14.07 129620 8.84

8239 3.27 24.39 225457 15.32

7030 2.76 20.81 190294 13.07

4265 1.65 12.62 113763 7.93

2000 0.77 5.92 53089 3.72

1072 0.42 3.17 28958 1.99

722 0.28 2.14 19305 1.34

452 0.19 1.34 13100 0.84

f1297 0.47 3.84 32405 2.41

3303 1.19 9.78 82047 6.14

5038 1.83 14.91 126173 9.37

6767 2.47 20.03 170299 12.58

8412 3.08 24.90 212357 15.64

7619 2.76 22.55 190294 14.16

6019 2.15 17.82 148236 11.19

4100 1.45 12.14 99973 7.62

2003 0.704 5.93 48539 3.72

1002 0.34 2.97 23442 1.86

868 0.31 2.57 21374 1.61

556 0.208 1.65 14341 1.03

g1013 0.389 3.00 26820 1.88

3095 1.17 9.16 80668 5.75

6556 2.485 19.41 171333 12.19

8239 3.123 24.39 215321 15.32

5000 1.851 14.80 127621 9.29

1700 0.621 5.03 42816 3.16

743 0.281 2.20 19374 1.38

3.17

6.15

8.27

12.24

17.89

21.40

12.52

6.05

2.87

1.66

1.93

3.13

6.64

12.60

18.52

32.07

9.63

3.76

2.49

1.86

1.50

1.64

2.05

2.97

6.05

11.68

14.19

23.20

13.07

4.21

1.99

1.59

2.55

7.41

17.55

#N/A

19.50

35.84

47.39

66.56

98.62

114.14

83.92

37.79

18.72

11.54

13.23

20.67

42.10

77.02

113.51

180.28

63.97

26.18

17.77

13.67

11.34

12.34

15.16

21.43

42.04

82.28

94.72

146.88

81.41

27.82

14.06

11.54

17.89

49.22

110.36

6.15

5.82

5.73

5.44

5.51

5.33

6.71

6.24

6.53

6.95

6.87

6.61

6.34

6.11

6.13

5.62

6.64

6.96

7.15

7.35

7.56

7.53

7.41

7.21

6.95

7.05

6.68

6.33

6.23

6.61

7.05

7.28

7.01

6.64

6.29

474.00

6.32

aBorda + screen (12/10/98).

bQuestionable.

°Borda + screen (12/12/98).

dBorda + screen (12/14/98).

+Orifice + screen (12/14/98; reverse of Borda + screen).

fOrifice without screen (12/14/98; reverse of Borda without screen).

gOrifice without screen (12/19/98; reverse of Borda without screen).

hN 1 points.

JN points.

JThrow out one point.

G1

0.0285

0.2466

0.3505

0.7740

0.6519

0.9753

0.1487

0.0059

0.0431

0.3964

0.3017

0.0832

0.0005

0.0434

0.0366

0.4877

0.1020

0.4104

0.6827

1.0651

1.5308

1.4576

1.1803

0.7837

0.3935

0.5276

0.1265

0.0001

0.0085

0.0817

0.5352

0.9154

0.4715

0.1039

0.0011

22.56

0.3048

0.3007

XEr_wl _Er_un

26.79

27.08

27.19

28.32

27.70

28.57

22.85

25.19

25.35

25.44

25.17

24.80

24.68

25.11

24.86

26.94

23.23

23.09

23.28

23.40

23.47

23.22

22.90

22.67

22.53

21.75

22.89

23.98

24.61

24.23

24.30

24.30

23.73

23.42

24.24

1938.03

25.84

G2

0.90

1.54

1.83

6.13

3.47

7.44

8.97

0.42

0.24

0.16

0.46

1.09

1.35

0.53

0.96

1.22

6.82

7.54

6.54

5.97

5.62

6.87

8.67

10.06

10.96

16.73

8.69

3.46

1.50

2.59

2.38

2.38

4.46

5.88

2.55

295.77

h4.00

J3.94

J3.648663

NASA/TM--2002-208914 42
TABLE 2._WISTED-TAPE WIDTH (w) AND THICKNESS (t) MEASUREMENTS

FOR 48 TWISTED TAPES a

[Sorted in ascending order (estimated).]

Tape

1

2

3

4

5

6

7

8

9

lO

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

Left-end measurements Right-end measurements

A= wt, A= wt, Thickness, Width,

• 2 . 2

m. m. (t), (w),

in. in.

Thickness, Width,

(t), (w),

in. in.

0.038 0.129

.038 .124

.038 .123

.039 .122

.039 .123

.039 .125

.04 .129

.04 .124

.041 .126

.042 .118

.124

.118

.129

.128

.124

.128

_r

.043 .121

.043 .121

.043 .119

.044 .119

.047 .123

.048 .145

.049 .13

.05 .124

.051 .125

.052 .124

.128

.123

.126

.125

.125

.124

,r .125

.053 .126

.053 .125

.054 .127

.123

.126

.123

_r .125

0.004902

.004712

.004674

.004758

.004797

.004875

.00516

.00496

.005166

.004956

.005208

.004956

.005418

.005376

.005208

.005376

.005203

.005203

.005117

.005236

.005781

.00696

.00637

.0062

.006375

.006448

.006656

.006396

.006552

.0065

.0065

.006448

.0065

.006678

.006625

.006858

.006642

.006804

.006642

.00675

0.004514

.004674

.004636

.004674

.005265

.004797

.00488

.00492

.005002

.005453

.00504

.005418

.005292

.005208

.004914

.005504

.005192

.00528

.005324

.005715

.00611

.0066

.0067

.0064

.006528

.0065

.006916

.006552

.0065

.007182

.00767

.00944

.007847

.008296

.008235

.007503

.007747

.007936

.008618

.008866

0.037

.038

.038

.038

.039

.039

.04

.04

.041

.041

.042

_v

.043

.044

.044

.044

.045

.047

.05

.05

.05

.051

.052

.059

.059

.059

.061

_v

.062

.062

.062

0.122

.123

.122

.123

.135

.123

.122

.123

.122

.133

.12

.129

.126

.124

.117

.128

.118

.12

.121

.127

.13

.132

.134

.128

.128

.125

.133

.126

.125

.133

.13

.16

.133

.136

.135

.123

.127

.128

.139

.143

a •

Average porosity, 0.6105. Corresponding average t (in.) = 0.0502 and w (in.) = 0.1273.

NASAfI'M--2002-208914 43
TABLE 2._CONCLUDED.

Left-end measurements Right-end measurements

A= wt, A= wt, Thickness, Width,

• 2 2

m. in. (t), (w),

in. in.

Tape

41

42

43

44

45

46

47

48

Totals

Average

Porosity

Thickness, Width,

(t), (w),

m. in.

0.055 0.132

.055 .126

.057 .125

.061 .127

.061 .132

.062 .118

.062 .126

.065 .128

2.334 6.01

0.04862

0.00726

.00693

.007125

.007747

.008052

.007316

.007812

.00832

0.008253

.008064

.009009

.00896

.00896

.008704

.00871

.008844

0.063

.063

.063

.064

.064

.064

.065

.066

0.131

.128

.143

.14

.14

.136

.134

.134

0.292508 0.319352 2.45 6.212

0.12521 0.05104 0.12942

0.6276 I 0.5934

aAverage porosity, 0.6105. Corresponding average t (in.) = 0.0502 and w (in.) = 0.1273.

H/D

3.62

4.34

5.8

8.0

22

Figure 1.--Single twisted tapes. (From Smithberg and

Landis, 1964.)

NAS A/TM------2002-208914 44
(b)

(c)

Figure 2.--Configurations of packed bed of twisted tapes. (a) Typical twisted-tape assembly. (b) Cross

section (end view). (c) Ideal packed bed in soccer ball configuration. (d) Test section. (e) Test section

screen. (f) Twisted tapes. (g) Prismatic bar model.

NASAfI'M--2002-208914 45
0

//

m

,.._ iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

NASA/TM---2002-208914 46
N-_8;B6rd_s_ree 48i odfi_e/noscreen

s E;Ho_o 34 s-L; Ho/O-22

K_h; Ho/O _ 5 _h; Ho/D 22

_ Dat 2i25 Date; 24

Date;_ 0;28

10 -3

10 =1 100 101 102 103 104 105

Ergun Reynolds number, XErgun

Figure &--Behavior for single twisted tape and 48 twisted tapes in packed bed relative to

Ergun model for laminar and turbulent-flow data.

NASAfI'M--2002-208914 47
_=_

0

0

0

= ==1_ lo -1

_. I_-_

kU

_J

O

 ̧'

 iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

10 .2

10 3 10 4

Ergun Reynolds number, XErgun

Figure 4.--Swirl velocity ratio correction (eq. (11)) applied to turbulent data.

10 5

Oil

flow

Alternative laser plane -,

F--

_/lirror-_ ,_/ I

/

/-- Laser plane

laser // (plane on which

/

/ / camera is focused)

/"

/

Light beam

r- Screen

/- Flow /

/ straighteners

Packed bed o1 //

twisted tapes-/_'_IV

Oil reservoir

Variable-speed pump J

camera

I

L_Flowmeter _--Pressure gauge

t_ _O-_tm oil filter

Figure &--Schematic of test facility.

NAS A/TM-------2002-208914 48
eO

"0

O_

0

a

e--

O

Q.

m

.Q

O

Q.

e--

O

e--

O

O

"1

iF_

NASAfI'M--2002-208914 49
Figure 7.--Transverse flow field at one diameter from packed-bed inlet.

1 0 2 _

O

E

O

E

ILl

iiiiiiiDat iiiIiamiii iaiiiiili i°isii ii iiiiii2iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii

E_gun model

o_

22

N-48i 0r ifiee/sere en _

100

10-1 100 101 102

Ergun Reynolds number, XErgun

Figure 8.--Laminar flow behavior for single twisted tape and packed bed of 48 twisted

tapes.

NAS A/TM-------2002-208914 50
10 2

101

t-

_.'" lo0

o

O

O

k_

O

10-1

eIll

10 -2

10 -3

10 0 101 10 2 10 3 10 4

Ergun Reynolds number, XErgun

Figure 9.--Velocity-corrected turbulent data and correlation functions.

105

10 2

g

S

0

0

,...

LLI

101

10 o

10 -1

........................ Ergun; X .............. _ ..................... 150_1 75X ...................................................................................................... !!! Erg_ Eig_ Erg_

10=2

10 -1 100 101 102 103 104

Ergun Reynolds number, XErgun

Figure lO.--Behavior of single twisted tape and packed bed of 48 twisted tapes relative to

Ergun model for laminar flow data.

NASA/I'M--2002-208914 51
=-

O

O

O

O

I.L

10 o

10 -1

10 -2

10 -3 I

10 -2

O

A

O

V

d3

Fiber Material

diameter,

mm

0.0389 Goat wool

0.0654 Blond hair

0.0735 Chinese hair

0.1504 Copper wire

7.156 Drill rod

C] _-_,1._- _ + O. = 3 ". _LU o / perpendicular , eq. (A98)

.... Eq. (A97a) --,

i Ill,Ill

10 -1 100 101 102 103 104

±Pd 2/_L Uo

Figure 11 .--Modified fit of Sullivan data (1942). (From fig. 2 of Hersh and Walker, 1980.)

NAS A/TM-------2002- 208914 52
=-

O

O

=o

O

O

.m

LI_

100 =--

10-1 __

10 -2

0

©

O

@

r3 Cotton wool

[] Camel hair

z5 Kapok

A Merino wool

<_ Cotton wool

O Rayon

V Down

T Glass wool

- _- Parallel

- \_ (eq. (A97a)) jO"

- \ _0 S*

\_ U _

e/3

+ 0.8 = ko (eq. (A95))

Glass wool

Glass wool and copper wire

Glass (fibers perpendicular to flow)

Glass (fibers parallel to flow)

\

_--Normal (eq. (A97b))

10 .3

10 -2 10 -1 100 101 102 103

_,Pd 2/p& Uo

Figure 12.--Modified fit of Davies data (1952). (From fig. 3 of Hersh and Walker,

1980.) ,&pd2 l _ko C______e_)2_.!_____P__d___2_ _

_r/Jo--] paramleN- 3 t_LU o J normal

NASA/I'M--2002-208914 53
O

_T

O

2.5

2.0

1.5

1.0

0.5

0.0

Values of YErgun

1%_ 150/XErgun + 1"75 0.16

I _ 150/XErgun + 4.5/XErgun

%%

-0.5 I J I Jl_l_l I J I Jl_l_l I J I Jl_l_l I J I Jl_l_l I J I Jl_l_l

100 101 102 103 104 105

I°gl0 XErgun

Figure 13.--Dependence of Ergun model including packed-sphere, turbulent-flow data of

Wentz and Thodos (1963).

1.0 m

0.5

0.0

= -0.5

O

O -1.0

-1.5-- Values°fYErgun0.16 l D ITL1"_'_--_______ _ -_B_ -- 150/XErgun + 4.5/XErgun Wentz and

-2.0 016

...... 0.03 (150/XErgun + 4.5/XErgun)J Thodos (1963)

E] XErgun/(Ue/U), Smithberg and Landis (1964); Koch (1958)

-2.5 I i I i IiIiI J i I i lllJl I i I i J Jill I i I J IillJ

101 102 103 104 105

log10 XErgun

Figure 14.--Relative magnitudes and slopes of flow parameters for packed beds of spheres

(Wentz and Thodos, 1963) compared with data for single twisted tape in tube (Smithberg

and Landis, 1964, and Koch, 1958).

NAS A/TM-------2002-208914 54
Form Approved REPORT DOCUMENTATION PAGE

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,

gathering and maintaining the data needed, and completing and review#lg the collection of information. Send corrlments regarding this burden estimate or any other aspect of this

collection of information, including suggestions for reducing this burden, to Washington Headquarters Services. Dhectorate for Information Operations and Reports, 1215 Jefferson

Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget Paperwork Reduction Project (0704-.0188), Washington, DC 20503.

1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3, REPORT TYPE AND DATES COVERED

November 2002 Technical Memorandum

4. TITLE AND SUBTITLE 5, FUNDING NUMBERS

Visu_ization of Flows in Packed Beds of Twisted Tapes

& AUTHOR(S)

R.C. Hendricks, M.J. Braun, D. Peloso, M.M. Athavale, and R.L. Mullen

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135 - 3191

WU-910-30-11-00

8. PERFORMING ORGANIZATION

REPORT NUMBER

E----1 1550-l

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

National Aeronautics and Space Administration

Washington, DC 20546- 0001 NASA TM------.2002- 208914

11. SUPPLEMENTARY NOTES

)

Prepared for the Second Pacific Symposium on Flow Visualization and Image Processing sponsored by the t acific Center of _lllermalFluids Engineering, Honolulu, Hawaii, May 16---19, 1999. R.C. Hendricks, NASA Glenn Research Center; M.J. Braun and D. Peloso,

University of Akron, Akron, Ohio 44325: M.M. Athavale CFD Research Corporation, Huntsville, Alabama 35805; and R.L. Mullen,

Case Western Reserve Universib_, Cleveland, Ohio 44106. A supplement CI) is enclosed which contains a videotape presentation of

flow visualizations in a packed bed of twisted tape. Responsible person, R.C. Hendricks, organization code 5000, 216---977---7507.

12a, DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category: 34 Distribution: Nonstandard

Available electronically at Ntp://gltrs.grc.n;_sa.gov

"l-his publication is ava_ilable from the NASA Center for AeroSpace In_brmation, 301-621-0390.

12b. DISTRNBUTION CODE

13. ABSTRACT (Maximum 200 words)

A videotape presentation of the flow field in a packed bed of 48 twisted tapes which can be simulated by very thin

virtual cylinders has been assembled. The indices of refraction of the oil and lhe Lucile twisted tapes were closely

matched, and the flow was seeded wilh magnesium oxide particles. Planar laser light projected the flow field in two

dimensions both along and transverse to the flow axis. The flow field was three dimensional and complex to describe,

yet the most prominent finding was flow threads. It appeared that mxial flow spiraled along either within the confines

of a virtual cylindrical boundary or within the exterior region, between the tangency points, of the virtual cylinders.

Random packing and bed voids created vortices and disrupted the laminar flow but minimized the entrance effects.

The flow-pressure drops in the packed bed fell below the Ergun model tbr porous-media flows. Single-twisted-tape

results of Smithberg and Landis (1964) were used to guide the analysis. In appendix A the results of several investigators are scaled to the Ergun model. Further investigations including different geometric configurations, computational

fluid dynamic (CFD) gridding, and analysis are required.

14. SUBJECT TERMS

Flow visualization

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

NSN 7540-01-280-5500

15. NUMBER OF PAGES

60

16. PRICE CODE

18, SECURITY CLASSiFiCATiON 19. SECURITY CLASSiFiCATiON 20. LiMiTATiON OF ABSTRACT

OF THIS PAGE OF ABSTRACT

Unclassified Uncl assifi ed

Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102