A Numerical Model for Radiative Heat Transfer

7/30/2019 A Numerical Model for Radiative Heat Transfer

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A Numerical Model for Radiative Heat Transfer Analysis inArbitrarily-Shaped Axisymmetric Enclosures with GaseousMedia

Edmundo M. Nunes([email protected] hat t an.edu)

and

Mohammad H.N. Naraghi([email protected])

Department of Mechanical E ngineeringManhattan College

Riverdale, NY 10471


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A b s t r a c t

A numerical model for evaluat ing t hermal radiat ive t ransport in irregularly-shaped axisymmet ric

enclosures conta ining a homogeneous, isotropically scat t ering medium is present ed. Based on t he

Discret e Exchange Factor (DEF ) method, exchange factors between arbit rarily-orient ed d ierential

surface/ volume ring elements are calculated using a simple approach. The present method is capable

of addressing blockage eect s produ ced by inner/ outer obst ruct ing bodies. T he r esults obtained

via t he current method are found t o b e in excellent agreement with exist ing solutions t o several

cylindrical media benchmark problems. The solutions t o several rocket -nozzle and plug-chamber

geometries are presented for a host of geometric conditions and optical thicknesses.

N o m e n c l a t u r e

a amplitude of cosine function in Eqs. (29) and (30)

dA surface ar ea of dierential ring element

dr radial t hickness of dierential volume r ing element

ds width of dierential surface ring element

ds s direct exchange factor between surface ring elements i and ji j

d ss surface-to-surface direct exchange factor matrix

D S S total exchange factor between surface ring elements i and ji j

D SS surface-to-surface total exchange factor matrix

ds v direct exchange factor between surface ring element i and volume ring element ji j

d sv surface-to-volume direct exchange factor matrix

D S V total exchange factor between surface ring element i and volume ring element ji j

D SV surface-to-volume total exchange factor matrix

dV volume of dierential ring element

dv s direct exchange factor between volume ring element i and surface ring element ji j

d v s volume-to-surface direct exchange factor matrix

D V S t otal exchange factor between volume r ing element i and surface ring element ji j

D V S volume-to-surface total exchange factor matrix

dv v direct exchange factor between volume ring elements i and ji j

d v v volume-to-volume direct exchange factor matrix

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D V V total exchange factor between volume ring elements i and ji j

D V V volume-to-volume total exchange factor matrix

dz axial t hickness of dierential volume ring element

E emissive power

I identity matrixK extinction coe cientt

N number of dierential ring elements

00q radiative heat ux

r radial coordinate

r radius of base of enclosureo

r position vector

R (z) local radius of inner obstructing axisymmetric bodyiR (z) local radius of outer obstructing axisymmetric bodyo

T temperature

w numerical integration weight factor

W weight factor matrix

z axial coordinate

G r e ek S y m b o ls

absorpt ion matr ix

angle between surface normal and vector connecting r and ri j

emissivity

, function de ned by Eq. (12)i j

, limit ing values of cos due to blockage by inner/ outer axisymmetric bodiesi o

tilt angle of surface with respect to z-axis

azimuth angle

reectivity

reectivity matrix

08 2 4 Stefan-Bolt zmann constant = 5.67051 10 W=(m K )

transmittance ( = K r )t o

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! scattering albedoo

Subs cr i pt s

i designates emitting ring elementj designates receiving ring element

max designat es maximum

min designates minimum

s designates surface

v designates volume

w designates wall

I n t r o d u c t i o n

Thermal radiative transport continues to emerge as an important energy transfer mechanism in

a wide variety of practical systems. For example, in high-pressure spacecraft engines, combust ion

product s can reach very high t emperatures, rendering radiation a signi cant mode of heat t rans-

fer. Moreover, thermal radiation heat transfer analyses are of primary interest in other specialized

computat ional arenas, such as gas t urbine, plug-chamber, fusion reactor, and crystal growth t her-

mal/ uid modeling. Many of t he a forementioned engineering systems ar e generally a xisymmetric

in shape, thereby reducing a computationally exhaustive 3-D analysis to a two-dimensional pro-

cedure. However, du e t o t he inherent complexities associat ed with radiat ive t ransfer calculat ions,

such as the long distance nature of radiation (solid angle integration), dependence on orientation

between participating elements, and functional dependence of radiative properties, radiative anal-

yses a re very oft en simpli ed. Consequently, simpli cation or neglect of radiat ive phenomenon in

t hermophysical models may inaccurately predict temperature and heat ux pro les. This is verymuch the issue in crystal growth modeling, where thermal gradients and interface shapes, which

direct ly aect cryst al quality and process sta bility, ar e sensitive t o radiat ive interact ions b etween

participating surfaces and/ or media. The aim of this paper is to present a model for systematically

evaluating radiative heat transfer in arbitrarily-shaped axisymmetric enclosures with participating

media.

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A close examination of the radiative heat tr ansfer lit erature reveals t hat t he analysis of ax-

isymmet ric domains is limited t o cylindrical enclosures, t he simplest of axisymmetric geomet ries.

Several numerical techniques have been implemented to solve for radiative transport within cylin-

drical geometries, including the Discrete-Ordinates (S-N) method, the Spherical Harmonics (P-N)

method, the Monte-Carlo method, the YIX quadrature, and t he zonal method. There are alsoseveral exact analytic solutions available for cylindrical media. Dua and Cheng [1] obt ained ana -

lytic expressions for nite and in nite cylinders with a non-scatt ering medium, whereas Crosbie and

Dougherty [2] examined absorbing, emit t ing, and scat t ering media within axially nite, but radially

in nite cylinders. The Discrete Ordinates method is used by Jamaluddin and Smith [3]. Although

t he Discrete Ordinat es met hod is accurat e and less memory intense t han other methods, it suers

from ray eect s [4]. Menguc and Viskant a [5] used t he Spherical Harmonics met hod (P 1 and P 3) t o

analyze cylindrical enclosures. T he P -N methods require a high order of approximation t o achieveaccurate results, especially in optically thin regions, and consume large amounts of computation

t ime and memory. Stewart and Cannon [6] used th e Monte Carlo method, which is exible, but

t ime-consuming and suers from inaccuracies due to st at ist ical error. Albeit highly accurate, t he

zonal method, used by Hottel and Saro m [7] to study radiative transfer in cylindrical furnaces, is

computationally intensive, requiring the evaluation of multiple spatial integrations for computing

exchange factors. All in all, t here exists a n eed for a comput at ionally e cient and exible scheme

for evaluating radiative transport in axisymmetric enclosures.

A numerical model has been developed based on the Discrete Exchange Factor (DEF) method

[8]. The DEF method, based on a point-to-point approach for r adiative analyses of enclosures,

has proven to be computationally advantageous over the zonal method (since multiple integrations

aren't necessary) and more accurate in one-dimensional systems [8]. The numerical results for two-

dimensional and three-dimensional systems are in excellent agreement with other methods [9, 10].

Since radiative exchange is computed between nodal points, the DEF method lends itself to grid

compat ibility with nite dierence/ element schemes for solving combined-mode heat t ransfer/ uid

ow pr oblems [11, 12]. Exchange factors between d ierent ial r ing element pairs a re comput ed by

generalizing Modest' s [13] model for view fact ors b etween d ierential ring elements on concentric

bodies t o give t he app ropriate DEF expressions. Blockage eect s p roduced by inner an d/ or out er

bodies are accounted for in the present formulation.

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M a t h e m a t ica l Fo r m u l at ion

Consider the arbitrarily-shaped axisymmetric enclosure shown in Fig. 1a. The enclosure, which

is comprised of an inner and out er surface, contains a radiatively part icipating medium. For sim-

plicity, all surfaces are assumed t o b e opaque, diuse and gray, and t he medium is homogeneous

and isotropically scatt ering. The DEF method computes radiat ive heat exchange between spatial

locations in enclosures by considering four avenues of direct radiative exchange: surface-to-surface,

surface-to-volume, volume-to-surface, and volume-to-volume. Integrating the nodal DEF equations,

given in [8], a bout t he circumferent ial direction for dierential elements i and j gives the following

direct exchange factor expressions between axisymmetric surface/ volume ring elements:

Z K jr r jt i jm a x2r ds cos cos ej j i jds s = d (1)i j j2 jr r jm i n i j

Z K jr r jt i jm a x2K r d r d z cos et j j j ids v = d (2)i j j2 jr r jm i n i j

Z K jr r jt i jm a xr ds cos ej j jdv s = d (3)i j j22 jr r jm i n i j

Z K jr r jt i jm a xK r dr dz et j j jdv v = d (4)i j j22 jr r jm i n i j

where symmetry with respect to the azimuth angle has been incorporated; r denotes the positioni

at which radiation is emitted; r denotes the position at which radiation is received; is the anglej

between the surface normal and the vector connecting r and r ; subscripts i and j denote emitt ingi j

and receiving elements, respectively; ds is t he width of the dierent ial surface ring element; andj

and are the minimum and maximum azimuth angles through which ring element j ism in m ax

seen from a point on ring element i , respectively.

The allowable range of is dictated by the orientation and relative position of the ring elementpair and blockage eects produced by inner and / or out er obstr ucting b odies. Details concerning

t he determination of the limit ing azimuth angles follow from Modest ' s [13] formulation and are

subsequently present ed. Geometric consideration of any ring element pair depicted in Fig. 1a.

reveals:

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2 2 2 2jr r j = r + r 2r r cos + (z z ) (5)i j i j j ii j

and for surface ring elements:

jr r j cos = (r r cos ) cos (z z ) sin (6)i j i i j i j i i

jr r j cos = (r cos r ) cos + (z z ) sin (7)i j j i j j j i j

where is the angle, resting in the r z plane, measured from the z-axis, in the direction ofk

increasing radius, onto the backside of the element k. For all inward facing surface ring elements,

=2 3=2, and for all outward facing ring elements, =2 =2. Combining Eqs. (1-7)

gives the resultant exchange factor expressions:

Z2 K jr r jt i jm a x2r r cos cos ds ( cos ) ( cos ) ei i j j i jjds s = d (8)i j j4 jr r jm i n i j

Z2 K jr r jt i jm a x 2K r cos dr dz ( cos ) et i j j ijds v = d (9)i j j3 jr r jm i n i j

Z K jr r jt i jm a x r r cos ds ( cos ) ei j j j jdv s = d (10)i j j32 jr r jm i n i j

Z K jr r jt i jm a xK r dr dz et j j jdv v = d (11)i j j22 jr r jm i n i j

where:r z z r z zi j i j i j = + t an ; = + t an (12)i i j jr r r rj j i i

The limiting angles and remain t o be determined. The limiting azimuth anglesm in m a xfor surface-to-surface exchange are governed by the con guration and orientation of both surface

ring elements and shadowing produced by inner and/ or outer blocking bodies. Radiative exchange

among surface ring elements inherently satis es the following condition:

cos 0 ; k = i; j (13)k

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Combining the geometric relations obtained in Eqs. (6) and (7) with the condition given above

results in an expression (functions and in Eq. (12)) for the cosine of potential limiting azimuthi j

angles for a surface ring element pair. If computed values and are in the interval of [-1,1], thei j

ring element pair is mutually fully or partial visible. For and outside of [-1,1], and dependingi j

on the orientation of the ring elements ( and ), the ring elements may be fully visible/ invisiblei jt o each ot her. Take, for insta nce, two horizonta l surface ring element s facing each other ( = =2i

and = 3=2). Both elements are fully visible to each other, yet values of and are both 1 ,j i j

outside t he int erval [-1,1]. Now, consider t wo horizont al surface ring elements facing in t he same

direction, with = = =2. Both elements clearly can not exchange radiant energy, yet andi j i

are, as in the example noted above, 1 . Values of cos and cos are systematicallyj m in m ax

determined by referring to Table 1, where potential azimuthal limiting factors are categorized for

speci c ranges of cos for each surface element . If, upon referring t o Table 1, cos < cos m in m a xfor a ring element pair, the exchange factor is taken as zero. It is important to note that the

determination of values for and for exchan ge among surface/ volume, volume/ surface,m in m a x

and volume/ volume ring elements is performed in a similar way. The dierence rests in recognizing

t hat volume ring elements h ave no orienta t ion a nd radiat e over a solid angle of 4. Thus, for volume

ring element j , the function is meaningless (since does not exist). Consequently, the selectionj j

of cos and cos from Table 1 becomes a simpli ed process.m in m a x

It is possible th at , in many instances, t he view between a r ing element pa ir is part ially obstru cted

1 1by an inner and/ or outer blocking body. Inner and outer blockage angles, cos and cos ,i o

are evaluated by projecting a line from a point on an emitting ring element (denoted by subscript

i ) around the periphery of the blocking body at an axial position z , such that z is between z andk k i

z . The intersection point between the receiving ring element (denoted by subscript j ) and shadowj

produced by the blocking body at z result in a potential minimum/ maximum azimuth angle. Thisk

procedure is repeated for several values of z and can be mathematically stated as:k

" #2 2 2 2 2 2R (z ) (z z ) r (z z ) r (z z )k j i j k k ii i j = max (14)i

2r r (z z ) (z z )i j k i j k z 2 (z ;z )i jk

" #2 2 2 2 2 2R (z ) (z z ) r (z z ) r (z z )k j i j k k io i j

= min (15)o2r r (z z ) (z z )i j k i j k z 2 (z ;z )i jk

For an inner cylindrical obstructing body, Eq. (15) reduces to:

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1" ! ! #2 2 2 2R R Ri i i = 1 1 (16)i 2 2r r r ri j i j

In an eort t o furt her clarify t he select ion process of and , consider again th e arbit rarily-m in m a x

shaped axisymmet ric enclosure shown in F ig. 1a. T he lines of view in t he r z plane, due to ringcon guration and blockage, from which ring elements dA and dV are seen from a point on dA arej j i

shown. These lines are projected onto the r plane via vertical and horizontal cuts through the

axisymmetric bodies, depicted in Fig. 1b. Labels a i denote circumferential positions on dA andj

dV , dividing them into viewed and unviewed portions. Note that the cosines of both and arej i j

both less than zero. The view from dA t o dA is aected by , , , and , producing unviewedi j i j i o

arcs ab; ac; hi ; and ae on dA . It is evident that cos = an d cos = . If there were noj m in o m a x i

obstructing bodies, then cos = and cos = 1. Radiat ive exchange bet ween element sm in j m a x

dA and dV is in uenced by , , and , yielding and as cos and cos , respectively.i j i i o o i m in m a x

A non-obstructed view between the ring elements would give values of and 1 for cos andi m in

cos . Figure 1c displays pr oject ed lines of sight for exchange b etween dV , dA , and dV . Arcsm ax i j j

ab; ad ; and fg represent unseen strips ofdA produced, r espectively, by pot ential limiting values ofj

cos of , , an d . The cosine of the minimum and maximum limiting angles are a n d .o j i j i

For radiat ive exchange between dV and dV , cos and cos are obtained from arcs ac andi j m in m ax

eg and are designated as and .o i

The exchange factors calculated from Eqs. (8-11) must satisfy the conservation of energy equa-

tions. The discretized form of these equations are given by:

N Ns vX Xw ds s + w ds v = 1 (17)s i j v i jj j

j = 1 j = 1

N Ns vX Xw dv s + w dv v = 1 (18)s i j v i jj j

j = 1 j = 1

where w (= ds ) and w (= dr dz ) are numerical integration weight factors for the discretizeds k v k kk k

conservation equations. These equations imply that radiant heat emitted from a surface or volume

element will be absorbed by other surface and volume elements in the enclosure. Since the exchange

factors are evaluated numerically via a 10-pt Gaussian integration routine, normalization procedures

are r equired so as t o obey t he conservat ion laws. In order t o account for direct radiative t ransfer

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between surface/ volume element s a nd diuse mult iple re ect ions at boundar ies a nd isotropic scat-

tering of radiation beams, total exchange factors are evaluated using an explicit matrix formulation

of the DEF method. Radiative a nalysis of multi-dimensional anisotropic scatt ering media can be

performed using the N-bounce or source function variations of the DEF method [14, 15]. However,

t he scope of this work is limited t o comput ing in an isot ropically scatt ering medium. Tot al exchangefactors are computed using the equations outlined below:

1 1D SS = [I fd ss + d sv ! W [I d v v ! W ] d v sgW ] o v o v s

1fd ss + d sv ! W [I d vv ! W ] d vsg (19)o v o v

1 1D SV = [I fd ss + d sv ! W [I d v v ! W ] d v sgW ] o v o v s

1d sv [I ! W d vv ] (1 ! ) (20)o v o

1D V S = [I d v v ! W ] d v s o v

1 1[I W fd ss + d sv ! W [I d v v ! W ] d vsg] (21)s o v o v

1 1D V V = [I d vv ! W ] d vv (1 ! ) + [I d v v ! W ] o v o o v

1 1d v s W [I fd ss + d sv ! W [I d vv ! W ] d v sgW ] s o v o v s

1d sv [I ! W d v v ] (1 ! ) (22)o v o

where D SS = [D S S ], D SV = [D S V ], DV S = [D V S ], D V V = [D V V ] are matrices of totali j i j i j i j

exchange factors between d ierential surface/ volume axisymmetr ic ring elements; d ss = [ds s ],i j

d sv = [ds v ], d vs = [dv s ], d v v = [dv v ] are mat rices of direct exchange fact ors b etween dier-i j i j i j

ential surface/ volume ring elements; W = [w ] and W = [w ] are diagonal matrices ofs s ;i i ;j v v ;i i ;j

numerical integration weight factors for surface/ volume ring elements, respectively; = [ ] andi;j

= [ ] are diagonal mat rices of re ect ivit ies and absorptivities for surface ring element s. Likei; j

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the direct exchange factors, the total exchange factors must obey conservation laws similar to those

given in Eqs. (17) and (18).

N Ns vX Xw D S S + w D S V = 1 (23)s i j v i jj j

j = 1 j = 1

N Ns vX Xw DV S + w D V V = 1 (24)s i j v i jj j

j = 1 j = 1

The radiative heat ux at each surface/ volume ring element is computed using the following

energy balance equation:

N Ns vX X00q = E w DS S E w D V S E (25)s ;i s;j j i s ;j v;j j i v;js ;i

j = 1 j = 1

N Ns vX X000q = E w D S V E w D V V E (26)v;i s;j j i s ;j v;j j i v;jv;i

j = 1 j = 1

where the gray emissive powers of the surface/ volume elements, E and E are de ned as:s ;i v;i

4E = T (27)s;i i s ;i

4E = 4K (1 ! ) T (28)v;i t o v;i

Results and Discussion

All numerical simulations were performed on a Silicon Graph ics workst at ion wit h a compu-

t ational t ime of a pproximately 20.3 seconds for a typical 20 10 mesh. All simulations were

additionally performed for a 10 5 and 40 20 mesh, each consuming 0.94 and 945.3 seconds

of computer time, respectively, t o ensure grid independence of the presented solutions. Severalbenchmark p roblems were identi ed an d solved in an eort t o validat e t he model pr esent ed here.

The rst benchmark problem concerns a cold, black cylinder with equivalent height and diameter,

encompassing a non-scattering, uniform temperature medium. Although Kassemi and Naraghi [16]

have solved t his problem using t he DEF method, we use it t o verify t he current formulat ion/ code in

t he absence of other benchmark dat a. Figure 2 presents t he dimenionsless radiat ive wall ux pro le

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for several solution techniques including an exact solution [1], spherical harmonics (P3) solution [5],

YIX solut ion [17], and t he present DEF solution. Th e DEF solution, similar t o t he YIX result s, is

in excellent agreement wit h t he exact solution for a ll levels of optical t hickness ( = 0.1, 1.0, and

5.0) shown. Th e P 3 solut ion, however, d eviat es from t he exact solution for low opt ical t hicknesses

( 1.0), overpredicting t he heat ux at t he cylinder ends.There are other cylindrical media benchmark data available for comparison. Wu and Fricker [18]

have compiled experimental wall heat ux data for a Delft furnace. The furnace, measuring 0.90 m

in diameter and 5 m in length, contains a gray, non-scattering medium with an average extinction

1coe cient of 0.3 m . Table 2 gives the non-uniform temperature distribution within the gas. The

furnace walls are gray and diuse with an emissivity of 0.8 and const ant t emperatu re of 425 K . The

comput ed heat ux distribut ions for t he Discret e-Ordinat es [3], YIX/ 16 [17], Finit e Volume [19],

P3 [5] and DEF methods are presented in Fig. 3. All of the solution methods, evidently predict thelocation of the peak heat ux accurately. In fact, most of the solutions are moderately close to one

another, with the exception of the P3 method, which seriously under-predicts the maximum heat

ux value. T here are, h owever, some note-worthy dierences b etween t he experimenta l dat a and

numerical solutions. These dierences are conceivably due t o t he presumption of a homogeneous

medium [17].

In high-pressure rocket -nozzles, comprehensive radiat ive analyses are di cult t o perform due

to the complexities introduced by shadowing eects at the throat. Hammad and Naraghi [20]

have developed a one-dimensional DE F scheme for evaluat ing radiant heat uxes in rocket -engine

geomet ries. In t heir formulat ion, t he exchange fact ors are comput ed b etween dierential volume

plates and/ or surface ring elements, warranted by the negligible variation of the combustion prod-

uct propert ies in t he radial and azimuthal direct ions. Results obta ined by t he 1-D [20] and 2-D

(present method) are compared in Fig. 4a for a typical rocket-engine with radial/ axial coordinates

and surface/ gas element t emperat ures speci ed in Table 3. T he nozzle wall is gray a nd diuse with

an emissivity of 0.95; t he combust ion gases are gray a nd non-scatt ering. T he entr ance/ exit sections

of the nozzle, shown at t he left-hand/ right-hand sides of t he comput ational mesh in Fig. 4b, are

assumed t o be black, with t emperat ures xed at t he rst surface segment and exit gas t empera-

1t ures, respect ively. Th e results show, t hat for ext inction coe cients of (K = 0.025 and 2.5 in ),t

both met hods are in good agreement with one another. However, t here are some discrepancies

1between the two methods for moderate extinction coe cients (K = 0.25 in ). It is likely t hatt

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t hese dierences can be att ributed to numerical inaccuracies associated with comput ing t he ex-

change factors for the 1-D method. These exchange factors require the evaluation of double, triple,

and quadruple integrals for computing surface-to-surface, surface-to-volume/ volume-to-surface, and

volume-to-volume exchange factors. For high optical thickness, the greatest contributor to the sur-

face ux comes from gas elements t hat are very close t o t he surface. T he computat ional error insuch a case is limited to a few volume-to-surface/ surface-to-volume exchange factor calculations. As

the medium becomes less optically dense, all four modes of radiative exchange become signi cant

between all of t he element s in t he domain, resulting in a relatively large build up of numerical

inaccuracies. As t he medium becomes more and more optically t hin, gas emission decreases sig-

ni cantly, and radiative t ransport among surfaces dominates, requiring t he evaluation of several

double integrals.

Given the conditions outlined in the rst benchmark problem, we have compiled the wall uxdistribution for nozzle shapes generated using the function below:

ir (z) = 1 a f(z) = 1 a 1 cos 2 (29)

N + 1w

where a denotes the amplitude of the cosine function; N is the total number of ring elementsw

comprising t he wall, and i is the wall element number, between 1 and N . Several simulations werew

performed for nozzle shapes with amplitudes ranging in value from 0.05 to 0.45 in steps of 0.05

(see Fig. 5a for a sample mesh layout ). For a 0.25, the wall ux pro les, given in Figs. 6 - 8 for

media with = 0.1, 1.0 and 5.0, show t hat the heat ux cont inually decreases from a peak value

at the throat (z=r = 1) to a minimum at the cylinder ends (z=r = 2). As t he amplitude of theo o

wall generating function is increased beyond 0.25, the throat becomes more narrow. This, in turn,

makes it di cult for surface elements in the vicinity of the throat (approximately 1.0 z=r 1.4)o

to view the entire domain. Consequently, the heat ux is relatively low at the throat and depending

on the value ofa, peaks at approximately 1.3 z=r 1.4 for media with low-to-moderate opticalo

thicknesses ( = 0.1 and 1.0). Similar tr ends are noted for optically t hick media ( = 5.0), due to

the relative low emission of radiant energy from volume elements in the throat region.

Radiative analyses of t he aforement ioned axisymmetric systems have required, at most, con-

sideration of blockage eect s produced by out er obstr ucting bodies. There are, however, many

cases of pract ical imp orta nce where shadowing eect s are caused by inner a xisymmetric b odies. In

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Czochralski growt h systems, for example, a cryst al pulled from t he melt part ially obst ruct s t he view

between t he furnace walls and melt surfaces. In t he area of spacecraft propulsion, t he signi cant

cost advantages of testing rocket nozzles by using plug-chambers is sparking the attention of many

researchers. Th e plug-chamber, used t o simulate t he ow-physics of rocket nozzles, consist s of a

diverging-converging inner plug cont ained within an outer cylindrical wall (see Fig. 5b). Severalanalyses were performed on various plug-chamber shapes generated using the following function:

ir (z) = b + a f(z) = b + a 1 cos 2 (30)

N + 1w

where b is the radius of the plug at the cylinder base and a is the amplitude of the cosine function.

Figures 9 - 11 give t he wall ux pro les for numerous plug-chambers shapes (see F ig. 5c for a

sample mesh layout) with a = 0.0 (inner obst ruct ing cylinder case) t o 0.40 in increment s of 0.05, b

= 0.1, and = 0.1, 1.0, and 5.0. The t hermal condit ions are equivalent to t hose prescribed in t he

rst benchmark problem. Th e gures exhibit t rends similar t o t hose report ed for t he rocket-nozzle

shapes. For a 0.20, the dimensionless wall ux decreases smoothly from a maximum at z=r =o

1 to a minimum at z=r = 2. As the value ofa is increased further, the heat ux at the middle ofo

the cylinder (z=r = 1) drops due t o t he dimishing visibility of this region by t he computat ionalo

domain, shifting the peak ux to a position given by 1.6 z=r 1.8. It should be noted that for o

= 5.0, t he wall ux d istribut ions for all of t he plug-chamber shap es examined converge in t he region

near the cylinder ends. This phenemonon is explainable in light of the short -distance a radiative

beam may travel in an optically dense medium and the xed con guration of the cylindrical wall.

C o n c l u d i n g R e m a r k s

The numerical evaluat ion of radiative heat t ransfer in ar bitr arily-shaped a xisymmetric enclo-

sures can be performed easily with the model presented in t his work. T he DEF method, t ogether

wit h a generalized view factor formulat ion between arbit rarily-orient ed dierent ial r ing elements

form the basis for systematically evaluating direct exchange factors between surface and/ or volume

elements. Since nodal placement is quit e arbit rary and exchange fact ors are evaluat ed in a direct

manner, the code is ideal for modeling radiative phenomena in axiymmetric, multi-phase thermal-

uid systems wit h moving and/ or deforming free-boun daries/ int erfaces. It is well-wort h ment ioning

that the exchange factors between arbitrarily-oriented ring elements are evaluated via one numer-

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ical int egration, making t he current method exible a nd computat ional e cient. A comparison of

the results obtained for several cylindrical media benchmark problems illustrate the accuracy of

t he method for a ll levels of optical t hickness. A comparison of 1-D and 2-D DEF solutions for a

rocket engine demonstrate good agreement between both methods for low/ high optical thicknesses.

However, at intermediat e opt ical conditions, t he 2-D met hod is found to be more accurate, sincemultiple integrations (two-to-four per exchange factor) between all elements comprising the domain

are n ot required. Th e solut ion t o several nozzle and plug-chamber shape geometries are included

to contribute to the benchmark literature.

Acknowledgments

This material is based upon work supported under a National Science Foundation Graduate

Research Fellowship and by ARPA/ AFOSR, as a part of The Consortium for Integrated Intelligent

Modeling, Design, and Control of Crystal Growth Processes.

References

[1] Dua, S.S. and Cheng, P., Multi-Dimensional Radiative Transfer in Non-Isothermal Cylindri-

cal Media Wit h Non-Isothermal Bounding Walls, International Journal of Heat and Mass

Transfer , Vol. 18, pp. 245-259, 1975.

[2] Crosbie, A.L. and Dougherty, R.L., Two-Dimensional Radiat ive Transfer in a Cylindrical

Geometry With Anisotropic Scattering, Journal of Quantitative Spectroscopy and Radiative

Transfer , Vol. 25, pp. 551-562, 1981.

[3] Jamaluddin, A.S. and Smith, P.J., Predicting Radiative Transfer in Axisymmetric Cylindrical

Enclosures Using the Discrete Ordinates Method, Combustion Science Technology, Vol. 62,

pp. 173-186, 1988.

[4] Lath rop, K.D., Ray E ect s in Discrete-Ordinat es Equat ions, Nuclear Science E ngineering,

Vol. 32, pp. 357-369, 1968.

[5] Menguc, M.P. and Viskanta, R., Radiative Heat Transfer in Axisymmetric Finite Cylindrical

Enclosures, Journal of Heat Transfer , Trans. ASME, Vol. 108, pp. 271-276, 1986.

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[6] Stewart, F.R. and Cannon, P., The Calculation of Radiative Heat Flux in a Cylindrical

Furnace Using the Monte-Carlo Method, International Journal of Heat and Mass Transfer ,

Vol. 14, pp. 245-261, 1971.

[7] Hott el, H.C. and Saro m, A.F., The Eect of Gas Flow Pa t terns on Radiative Transfer in

Cylindrical Furnaces, International Journal of Heat and Mass Transfer , Vol. 8, pp. 1153-1169,

1965.

[8] Naraghi, M.H.N., Chung, B.T.F., and Litkouhi, B., A Continuous Exchange Factor Method

for Radiative Exchange in Enclosures With Participating Media, Journal of Heat Transfer,

Vol. 110, No. 2, pp. 456-462, 1988.

[9] Naraghi, M.H.N. a nd Kassemi, M., Radiative Transfer in Rect angular Enclosures: A Dis-

cretized Exchange Factor Solution, Journal of Heat Transfer, Trans. ASME , Vol. 111, No. 4,

pp. 1117-1119, 1989.

[10] Naraghi, M.H.N. and Litkouhi, B., Discrete-Exchange Factor Solution of Radiat ive Heat

Transfer in Three-Dimensional Enclosures, Radiation Heat Transfer: Fundamentals and Ap-

plications, ASME publication HTD-Vol. 137, pp.133-140, 1989.

[11] Saltiel, C. and Naraghi, M.H.N., Analysis of Radiative Heat Transfer in Participating Media

Using Arbitrar y Nodal Dist ribut ion, Numerical Heat Transfer Journal, Part B: Fundamentals,

pp. 227-243, 1990.

[12] Saltiel, C. and Naraghi, M.H.N., Combined-Mode Heat Transfer in Radiatively Participating

Media Using th e Discret e Exchange Factor Met hod wit h F init e Elements, Heat Transfer 1990,

Hemisphere Publishing Corporation, Vol. 6, pp. 391-396, 1990.

[13] Modest, M.F., Radiat ive Shape Fact ors Between Dierent ial Ring Elements on Concentric

Axisymmetric Bodies, AIAA Journal of Thermophysics and Heat Transfer , Vol. 2, No. 1, pp.

86-88, 1988.

[14] Naraghi, M.H.N. and Huan, J., An N-Bounce Method for Analysis of Radiative Transfer in

Enclosures with Anisotropically Scattering Media, Journal of Heat Transfer, Vol. 113, No. 3,

pp. 774-777, 1991.

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[15] Huan, J. and Naraghi, M.H.N., Source Function Approach for Radiative Heat Transfer Anal-

ysis, AIAA Journal of Thermophysics and Heat Transfer , Vol. 6, No. 3, pp. 568-571, 1991.

[16] Kassemi, K. and Naraghi, M.H.N., Application of Discrete Exchange Factor Method to Com-

bined Heat Transfer Problems in Cylindrical Media, HT-6D: Transport Phenomena in Man-

ufacturing and Materials Processing{II, Proc. 1996 ASME Winter Annual Meeting, Atlanta,

Georgia, 1996.

[17] Hsu, P.-F. and Ku, J.C., Radiative Heat Transfer in Finite Cylindrical Enclosures with Non-

Homogeneous Participating Media, AIAA Journal of Thermophysics and Heat Transfer , Vol.

8, No. 3, pp. 434-440, 1994.

[18] Wu, H.L. and Fricker, M., The Characteristics of Swirl-Stabilized Natural Gas Flames{Part

2: T he Behavior of Swirling J et Flames in a Narrow Cylindrical Furnace, Journal Inst. of

Fuel, Vol. 49, pp. 144-151, 1976.

[19] Chui, E.H., Raithby, G.D., and Hughes, P.M.J., Prediction of Radiative Transfer in Cylindri-

cal Enclosures with the Finite Volume Method, AIAA Journal of Thermophysics and Heat

Transfer , Vol. 6, No. 4, pp. 605-611, 1992.

[20] Hammad, K.J. and Naraghi, M.H.N., Exchange Factor Model for Radiative Heat Transfer

Analysis in Rocket Engines, AIAA Journal of Thermophysics and Heat Transfer , Vol. 5, No.

3, pp. 327-334, 1991.

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Table 1: Limiting values for the cosine of the azimuth angle.

cos 0 cos 0i icos 0 cos = min( ; 1) cos = min( ; ; 1)j m in o m in i o

cos = max(; ; ; 1) cos = max( ; ; 1)m ax i j i m a x j icos 0 cos = min( ; ; 1) cos = min(; ; ; 1)j m in j o m in i j o

cos = max( ; ; 1) cos = max( ; 1)m ax i i m a x i

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Table 2: Temperature distribution within the Delft furnace [18].

z, (m) T(r = 0:075m), (K ) T (r = 0:225m), (K ) T(r = 0:375m), (K )0.15 1470 1120 870

0.45 1600 1320 1070

0.75 1620 1470 13601.05 1610 1550 1370

1.35 1580 1520 1350

1.65 1520 1470 1320

1.95 1470 1410 1280

2.25 1410 1360 1250

2.55 1350 1310 1210

2.85 1310 1260 1170

3.15 1270 1230 1150

3.45 1240 1200 1110

3.75 1200 1160 1090

4.05 1170 1130 1080

4.35 1140 1100 1070

4.65 1110 1080 1060

4.95 1080 1070 1060

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Table 3: Nozzle radii and surface/ gas temperature distribution along axial direction [20].

z, (in ) r , (in ) T , (R ) T , (R)s v-7.572 2.4199 1250.0 6708.6

-6.843 2.3824 1244.4 6708.3

-6.115 2.3374 1237.1 6707.5

-5.386 2.2802 1222.0 6706.0

-4.658 2.1951 1208.8 6703.6

-3.929 2.0821 1205.1 6699.9

-3.200 1.9446 1131.2 6692.6

-2.472 1.7924 1062.9 6680.2

-1.743 1.6380 1078.1 6660.8

-1.015 1.4833 1106.5 6621.9

-0.286 1.3743 1105.8 6509.7

0.443 1.4056 908.8 6150.6

1.171 1.5670 896.8 5885.3

1.900 1.7629 926.0 5692.7

2.628 1.9576 871.3 5549.5

3.357 2.1534 803.3 5415.14.086 2.3481 736.2 5306.3

4.814 2.5437 669.3 5203.9

5.543 2.7408 579.1 5116.1

6.271 2.9422 479.6 5034.3

7.000 3.0993 380.0 4952.5

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F i g u r e C a p t i o n s

Figure 1: (a) Two-dimensional view of axisymmetric b odies in r z plane; (b) Projection of view

from surface ring element dA to ring elements dA and dV in r plane, and (c) Projection ofi j jview from volume ring element dV to ring elements dA and dV in r plane.i j j

Figure 2: A comparison of dimensionless wall ux p ro les obta ined from DEF , E xact, YIX, and P 3

methods for cylindrical enclosures with = 0.1, 1.0 and 5.0.

Figure 3: A comparison of wall heat ux dist ributions predicted using DEF , YIX, Finite Volume,

S4 and P3 methods with experimental Delft furnace data [18].

Figure 4: (a) A comparison of dimensionless wall uxes based on 1-D [20] and 2-D (present) DEF

1methods for K = 0.025, 0.25 and 2.5 in and (b) Computational mesh of rocket engine.t

Figure 5: (a) Mesh layout for a nozzle with a = 0.25; (b) Geometric con guration of a sample

plug-chamber, and (c) Mesh layout for a plug-chamber with a = 0.20.

Figure 6: Dimensionless wall ux pro les of nozzles wit h a ranging in value from 0.05 to 0.45 in

steps of 0.05 for = 0.1.





Figure 9: Dimensionless wall ux pr o les of plug-chamb ers wit h a ranging in value from 0.0 to 0.40

in steps of 0.05 for = 0.1.

Figure 10: Dimensionless wall ux pro les of plug-chambers with a ranging in value from 0.0 to

0.40 in steps of 0.05 for = 1.0.

Figure 11: Dimensionless wall ux pro les of plug-chambers with a ranging in value from 0.0 to

0.40 in steps of 0.05 for = 5.0.

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A Numerical Model for Radiative Heat Transfer