Research ArticleNumerical Study on Similarity of Plumersquos Infrared Radiationfrom Reduced Scaling Solid Rocket

Xiaoying Zhang and Rui Li

South China University of Technology Wushan Road No 381 Guangzhou 510641 China

Correspondence should be addressed to Xiaoying Zhang zxy1119scuteducn

Received 7 January 2015 Revised 22 March 2015 Accepted 22 March 2015

Academic Editor Yogesh Jaluria

Copyright copy 2015 X Zhang and R Li This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Similarity of plume radiation between reduced scaling solid rocket models and full scale ones in ground conditions has been takenfor investigation Flow and radiation of plume from solid rockets with scaling ratio from 01 to 1 have been computedThe radiativetransfer equation (RTE) is solved by the finite volume method (FVM) in infrared band 2sim6 120583m The spectral characteristics ofplume gases have been calculated with the weighted-sum-of-gray-gas (WSGG) model and those of the Al

2O3particles have been

solved by the Mie scattering model Our research shows that with the decreasing scaling ratio of the rocket engine the radiationintensity of the plume decreases with 15sim25 power of the scaling ratio The infrared radiation of the plume gases shows a strongspectral dependency while that of the Al

2O3particles shows grey property Spectral radiation intensity of the high temperature core

of the solid rocket plume increases greatly in the peak absorption spectrum of plume gases Al2O3particle is the major radiation

composition in the rocket plume whose scattering coefficient is much larger than its absorption coefficientThere is good similaritybetween spectral variations of plumes from different scaling solid rocketsThe directional plume radiation rises with the increasingazimuth angle

1 Introduction

As for base heating and thermopneumatic property of thesolid rocket plume radiation has been the focus of theinvestigation in thermal protection design of solid rockets inpast decades [1] Much related experimental and theoreticalstudy has been carried out to study radiation characteristicsof the solid rocket plume As a limitation of ground test mostexperiment study of rocket plume radiation used reducedscaling model and also some of the theoretical studies usedscalingmodel Considering the application of research resultswith reduced scaling models in radiation knowledge andthermal protecting design of full scale rockets similaritybetween plume radiation of reduced scaling model and fullscale rocket must be obtained However to the authorsrsquo bestknowledge there has not been experimental study about thatproblem published till now Only several numerical analysisstudies can be found out

Rozanov and Lyapustin had studied a new integral formof similarity conditions to the error analysis of truncationtechniques for the forward peak scattering and deducedthe integral similarity method that is Delta-M [2] Jun and

Wenbing had studied the self-similarity of transmittancedepth and radiation energy in a cool medium with chang-ing time boundary temperature and medium density [3]Research of Duracz and Mccormick was focused on theratio of two similarity parameters radiation intensity andirradiance and the influence of the single-scattering albedoand asymmetric coefficients within weakly absorbing clouds[4] Mitrescu and Stephens had studied a new approach fordetermining the scaling parameter of the radiative transferequation and proposed that the key assumption regarding theangular dependence of the radiative field is essential [5] Brilet al had studied the similarities between temperature fieldand concentration field in the near nozzle area this studyalso proposed a method for determining a nondimensionalradiance intensity with the outlet parameter radiation wave-length and temperature gradient of absorptivity [6]

Calculation of solid rocket plume radiation is highly com-plicated as the RTE is a high ordered nonlinear equation withdifferential term and integral terms considering the strongspectral properties of plume gases and radiative absorptionand scattering of groups of Al

2O3particles with different

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 627351 11 pageshttpdxdoiorg1011552015627351

2 Journal of Applied Mathematics

temperatures and diameters The plume gases consist ofseveral polarity gases H

2O CO

2 CO HCL and OH whose

infrared radiative spectrum is made up with ten thousandsof lines [7] The common detailed calculating method for gasradiation uses a band model [8 9] or the weighted-sum-of-gray-gas (WSGG) model [10] The band model has a higherprecision for nonhomogeneous gases but is somewhat timelasting for large volume gases and is not very suitable forsolid rocket plume gases The WSGG model always showsa compromise of precision and calculation time which hasbeen widely used to calculate radiation of clouds or combus-tion gases As for spectral property of Al

2O3particles effects

of particle temperature concentrations and diameters areneeded to be considered [11 12] and theMie scattering theoryfor calculating particle radiation can also be used for Al

2O3

particles [13] Many numerical solution methods have beendeveloped to calculate radiation in nonhomogenous absorp-tiveemissivescatteringmediumby solving RTEnumericallyincluding Monte Carlo (MC) methods [14] the streamingmethod (SM) [15] discrete-ordinate method (DOM) [16]and finite volume methods (FVM) [17] To improve thesolution accuracy with less execution time the FVM hasoften been used to calculate the spectral radiation of thehigh temperature two-phase plume with highly expandingvolume

To study the similarity of infrared radiation from plumeswith different scaling ratio the Trident D5 solid rocket anda series of reduced scaling models with similar geometryhave been taken for investigation CFD code has been usedto compute the axial symmetric flowing field inside therocket engine and in plume of the full scaled rocket andreduced scaling model with same combustion chamberpressure and plumersquos flowing Mach number The spectralabsorption coefficient of plume gases H

2O CO

2 CO HCL

and OH will be computed with WSGG model Eight groupsof Al2O3particles with different diameters between 2 and

16 120583mhave been taken into considerationTheMie scatteringmodel has been used to compute the spectral absorptionscattering coefficient and phase function of Al

2O3particles

To calculate the spectral radiance of the plume a finitevolume method has been used to solve the RTE Integratingspectral radiance on exterior surface of plume volume in onedirection will give the plume radiation intensity Ratio ofradiation intensity between reduced scaling model and fullscaled one has been computed to study the similarity rules ofplume radiation

2 Calculation of Plume Flowing Field

The Trident D5 solid rocket used a composite propellantNEPE there is 10 weight of Al in the composite propellantPlumersquos flowing data of the full scaled rocket and reducedscalingmodels are calculated with the same input data for therocket nozzle pressure is 9Mpa and temperature is 3750KInput concentrations of major species are 119891OH = 0192 119891H

2

=0299 119891CO = 0027 119891CO

2

= 0146 and 119891H2O = 00008 on the

inlet of nozzle The full scaled rocket has a nozzle with 155mlength 035m diameter and 97 ratio of area expansionThe reduced scaling model has the same geometry but with

a minimized diameter and length To obtain the flowing dataof plume the CEA program has been adopted to compute thechemical-equilibrium compositions of the propellants andthe inlet parameters

Then CFD code is used to compute the united flowingin rocket nozzle and plume The time-marching method andthe advection upstream splitting method (AUSM) have beenused for numerical discretization of the 2D axis symmetricN-S equationsThe 119896-120576 turbulence model is adopted to simulateblending of the plume and atmosphere A finite rate chemicalreaction model with 12 components and 17 reactions hasbeen taken for consideration the detailed reaction equationsand related coefficients can be seen in [18] For the Al

2O3

particles the Lagrangian particle trajectory model has beenused to simulate exchange of energy andmomentumbetweenparticles and plume gases The distribution of the particlediameter is determined by Braithwaitersquos proposed functions[19]

The plumersquos flowing data of the full scaled Trident D5rocket and the 9 reduced scalingmodels have been calculatedin our study Only two groups of result are given in Figure 1One is for the full scaled rocket and the other is for the 05scaling model Considering the axial symmetric character-istics of the plumersquos flowing field only flowing field map onthe upper 119909-119911 plane has been plotted Figure 1 shows pressureand temperature of plume gases and volume fractions of the 3major components H

2O CO

2 and CO Volume fractions of

HCL and OH molecule have not been shown here as theirfractions are very small less than 005 Temperature andnumber concentration of the 8 groups of Al

2O3particles have

been calculated in our study only results of 3 groups withdiameters being 6 120583m 8 120583m and 10 120583m are given here

One can see from the figure that volume of the plumeexpands rapidly after the rocket nozzle making the diameterof the high temperature core at 119909 = 20m about 5 times that ofthe nozzle exit All maps of the flowing map of the 05 scalingrocket model have close similarity with the full scaled rocketwith similar parameter distributions and a close data rangeOne can also find that the high temperature areas of gases andAl2O3particles are a continuous central stripe in the plume

but the dense concentration area of Al2O3particles number

is quite discrete Comparing volume fraction of CO2and CO

it can be found that the high volume fraction area of CO2is

in the radial medium ring at some axial distance after thenozzle exit while that of CO is in the central thin stripe closebehind the nozzle exit For H

2O the high volume fraction

area is continuous in the central stripe of the plume

3 Calculation Method of Plume Radiation

A widely used equation of radiation transport in a nonho-mogeneous absorptiveemissivescattering medium can bewritten as variation of radiance 119894

1015840

120582(

S)while passing a distance119889S along the path S in the medium [20 21]

119889i1015840120582(S)

119889S= minus (Σ120581

119899+ 120590) i1015840120582(S) + (Σ120581

119899) i1015840120582119887

+

120590

4120587

int

4120587

i1015840120582(S1015840)Φ (S S1015840) 119889120596

1015840

(1)

Journal of Applied Mathematics 3

fCO2

1 32 4 6 85 7 9

2 4 6 8 10 12 1614 18

fH2O

0 84 12 2016 2824

tg

420

5 10 15

z

x

005 01 015 02 025

fCO

0

2E+10

3E+10

1E+10

4E+10

5E+10

6E+10

7E+10

500 1000 1500 2000 2500

tp3

0

12E

+10

6E+09

24E+10

18E+10

36E+10

3E+10

48E+10

42E+10

0

4E+9

6E+9

2E+9

8E+9

12E+10

14E+10

1E+10

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

prg2 4 6 8 10 12 14 16

times104times102 times10minus2

times10minus2

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

am3

am4 am5

(a) Plume flowing data of 05 scaling sized model

2 4 6 8 10 12 1614 18

fH2O

005 01 015 02 025

fCO

1E+10

2E+10

4E+10

3E+10

5E+10

0

1E+10

2E+10

4E+10

3E+10

5E+10

x

prg2 4 6 8 10 12 14 16

times104

tgtimes102

times10minus2

fCO2

1 32 4 6 85 7 9times10minus2

z

x

am3

500 1000 1500 2000 2500

tp3

0

4E+9

8E+9

12E+10

16E+10

2E+10

am4am5

2

1

02 4 6 8

z

2

1

02 4 6 8

z

x

2

1

02 4 6 8

x

z

x

2

1

02 4 6 8

z

2

1

02 4 6 8

z

x

2

1

02 4 6 8

x

z

x

2

1

02 4 6 8

z

2

1

02 4 6 8

z

x

2

1

02 4 6 8

0 4 8 12 16 24 28

(b) Plume flowing data of full scaled rocket

Figure 1 Plume flowing data of the 05 scaling model and full scaled solid rocket

4 Journal of Applied Mathematics

2

0

z(m

)

5 10 15 20

x (m)

Figure 2 Grids for radiance calculation in the plume

where the subscript ldquo120582rdquo means spectral variable and thesuperscript ldquo1015840rdquo means directional variable The symbol Σ120581

119899

is the summed absorption coefficient of plume gases andAl2O3particles and 120590 is the summed scattering coefficient

of 8 groups of Al2O3particles which are simply summed

up mathematically The symbol S1015840 means direction of theincident radiation 1205961015840 means the solid angle in S1015840 directionand Φ(S S1015840) means the scattering phase function betweenS and S1015840 The above symbols 120581

119899 120590 and Φ are all spectral

variables while the notations ldquo120582rdquo are omitted in (1) forabbreviation

According to finite volume method (1) can be solvedto compute spectral radiance of plume by integrating theequation on a control volume Considering the plume geom-etry the cylindrical control volumes had been firstly usedfor solving (1) but it is found that numerical convergence ofthe developed coefficient matrix is very poor being as thearea difference between the two radial opposite faces of thecontrol volume For that reason the cuboid control volumehas finally been chosen in our study The new computationdomain for radiation computation has the same length withthe flowing domain in Figure 1 but it has a square formend with side length being equal to the outer radius of theplume The precision of plume radiance solution is mainlyaffected by flowing data temperature volume fraction ornumber concentration of plume gases and Al

2O3particles

Those flowing data do not change dramatically in the plumeas seen in Figure 1 so the mesh size for radiance solution canbe much larger than that used in CFD solution For plume ofthe full scaled solid rocket the meshing plot on lengthwisesection of the computing domain is shown in Figure 2 Thedivided grids number is 80 times 20 in the upper 119909-119911 planeFlowing data of control volume inside the plume takes thearithmetic average of data on CFD grids in the volume Thatof control volume outside the plume is set to zero In thatcase all control volumes outside the plumewould not actuallyinfluence the result of plume radiance but the integral RTEequation is uniform for all the control volumes and solutionconvergence can be improved

Spectral radiance of each control volume is consideredin a series of directions every direction is correspondingto a set of angle zenith angle and azimuth angle Thezenith angle 120579 is defined as the angle between the plumecentral line (119909-axis) and the direction vector and azimuthangle 120593 is defined as angle between the projected directionvectors on 119909-119911 coordinate plane and the 119911 axis For radianceon control volume 119875 in direction S1015840 integrating (1) withGaussian integral method radiance of 119875 can be related toradiance of its neighbored control volumes in that directionThe detailed derivation procedure can be found in [19] onlythe final integration equation is given here as follows

f

w

NF

R

S

EW P

rs

n

x

y

e

z

Figure 3 A control volume and the 6 neighbors

max [Δ119860

119908119863

119908 0] i1015840119875+ max [Δ119860

119890119863

119890 0] i1015840119875

+ max [Δ119860

119904119863

119904 0] i1015840119875+ max [Δ119860

119899119863

119899 0] i1015840119875

+ max [Δ119860

119903119863

119903 0] i1015840119875+ max [Δ119860

119891119863

119891 0] i1015840119875

+ (Σ120581

119899+ 120590)Δ119881

119875Δ120596

1015840i1015840119875

= max [minusΔ119860

119908119863

119908 0] i1015840119882

+ max [minusΔ119860

119890119863

119890 0] i1015840119864

+ max [minusΔ119860

119904119863

119904 0] i1015840119878+ max [minusΔ119860

119899119863

119899 0] i1015840119873

+ max [minusΔ119860

119903119863

119903 0] i1015840119877+ max [minusΔ119860

119891119863

119891 0] i1015840119865

+ [Σ (120581

119899) i1015840120582119887

+

120590

4120587

Σ (i1015840119875(S) Φ (S S1015840) Δ120596)]Δ119881

119875Δ120596

1015840

119863

119896= int

Δ1205961015840

(S sdot n119896) 119889120596

1015840 119896 = 119908 119890 119899 119904 119903 119891

Φ (S S1015840) =

int

Δ1205961015840int

Δ120596Φ(S S1015840) 119889120596

1015840119889120596

Δ120596

1015840Δ120596

Δ120596

1015840= int

120593119897+

120593119897minus

int

120579119897+

120579119897minus

sin 120579 119889120579 119889120593

(2)

where subscripts 119882 119864 119873 119878 119877 and 119865 mean the variables ofthe neighbored control volume and 119908 119890 119899 119904 119903 and 119891 meanthe variables on interface of neighboring volumes as shownin Figure 3 The symbol Δ119860

119896is the area of the interface n

119896is

external normal vector of interface and Δ119881

119875is the volume of

control volumeThe upper equations can be also written as the following

simplified form

119886

119875i1015840119875

= 119886

119882i1015840119882

+ 119886

119864i1015840119864+ 119886

119878i1015840119878+ 119886

119873i1015840119873

+ 119886

119877i1015840119877+ 119886

119865i1015840119865+ b119875

119886

119875= Σmax (Δ119860

119896119863

119896 0)

Journal of Applied Mathematics 5

+ (Σ120581

119899+ 120590 minus

120590

4120587

Φ (S1015840 S1015840) Δ120596

1015840)Δ119881

119875Δ120596

1015840

b119875

= [(Σ120581

119899) i1015840120582119887

+

120590

4120587

sum

S =S1015840i1015840119875(S) Φ (S S1015840) Δ120596]Δ119881

119875Δ120596

1015840

(3)

The compound integral RTE equations of all control volumeshave the form of vector equations

[A] [I] = [B] (4)

Equation (4) is a nonlinear system of equations as variableb119875in the right side is actually unknown and needs to be

determined by radiance of control volume 119875 in all the otherdirections S1015840 So the solution of (4) could not be completedwith any iteration algorithm in one loop A cyclic iterationalgorithm with modification of b

119875has been proposed in this

study In each cycle b119875is firstly calculated with the radiance

computed from the last cycle Then a Gauss iteration algo-rithm will be used to solve vector equations (4) to computea new group of radiance for each control volume The cycliciteration will go on till the following convergence limitationis met

max i119899119875minus i119899minus1119875

le 120576 (5)

Considering the symmetric characteristics of geometry andflowing data of the plume radiance of plume is also sym-metric So we do not need to solve the vector equations ofRTE on all control volumes in the plume but only thosecontrol volumes with centers on 119909-119911 coordinate plane havebeen chosen for calculation While radiance of neighboredcontrol volumes is also required in (4) those control volumesare not centering on the 119909-119911 coordinate plane and a radiancetransformation algorithm on symmetric control volumes hasbeen proposed in our study to calculate the radiance of othercontrol volumes in the plume The radiance transformationalgorithm is derived based on rotation rules of vectors seenin Figure 4

If the radiance of control volume 119875 in direction S1is

i1015840119875(S1) the radiance of control volume 119874 in direction S

2is

equal to i1015840119875(S1)when the following vector equation is satisfied

p2= S2+ r2 (6)

4 Calculating Radiation Intensity fromthe Plume Surface

In the cases of studying base heating or heat protection of thesolid rocket spectral radiation intensity on the whole surfaceof the plume is concerned Spectral radiation intensity is alsoa directional variable which is the integration of spectralradiance i1015840

120582(S) on surface of bounding control volumes

in direction S If radiation intensity in some waveband

z

PQ

y

1198511

1198261

1198262

1198262

1198491

11984921198512

120575120575

120579

120579

Figure 4 Radiance transformation on symmetric control volumes

is concerned as for the 2sim6120583m band integration on thewavelength is also needed

119868

120582= sum

119895119860

i1015840120582(S) eS sdot n

Δ119860Δ119860 eS sdot n

Δ119860gt 0

119868 = sum

119895120582

sum

119895119860

i1015840 (S) eS sdot nΔ119860

Δ119860Δ120582

(7)

where 119895

119860is number of control volume surfaces on boundary

of the plume eS is unit vector of the radiance direction andnΔ119860

is unit vector of external normal on the surface

5 Calculation of Spectral Property ofPlume Gases and Al2O3 Particles

The3major radiation gases of the solid rocket plume areH2O

CO2 and CO To calculate the spectral absorption coefficient

of each plume gas the standard absorption coefficient 120581120582STP

at several temperatures and 1 atm pressure in [20 21] has beenused For plume gas with pressure being 119875 and temperaturebeing 119879 absorption coefficient can be modified from 120581

120582STPas follows

120581

120582=

119875

10

5

273

119879

120581

120582STP (8)

Absorption coefficient of the plume gases mixture is calcu-lated with the WSGG model and the weight is the volumefraction of each gas If the volume fraction of one gas is 119891then the absorption coefficient of the gases mixture can bewritten with the WSGG model as

120581

120582119892= 120581

120582H2O119891H

2O + 120581

120582CO2

119891CO2

+ 120581

120582CO119891CO (9)

The calculated absorption coefficient for plume gases H2O

CO2 and CO and the gases mixture in center of front end of

the plume have been shown in Figure 5 One can find thatabsorption coefficient of CO

2and CO is comparatively much

larger than that of CO is 42sim55 120583m the peak spectrum ofCO2is 42sim5 120583m and absorption coefficient of H

2O is much

smaller Absorption coefficient of the gases mixture appearsas an accumulated effect of the three gases

6 Journal of Applied Mathematics

0

1

2

3

4

5

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

kH2OkCO2kCO

kHCLkgases

Figure 5 Calculated absorption coefficient of plume gases withWSGG model

6 Calculation of the Spectral Property ofAl2O3 Particles

In the calculation of spectral properties of Al2O3particles

with Mie scattering model particle size number concen-tration and complex refractive index must be determinedfirstly Asmentioned above size andnumber concentration ofAl2O3particles have been calculated in plume flowing com-

putation The complex refractive index of Al2O3particles

m = 119899 + i119896 takes formulas recommended by Reed et al [12]

119896 = 466 times 10

minus4120582

133119879

15 exp [minus

29420

119879

]

119899 = 175 cos (6120582) (10)

For a single Al2O3particle spectral properties including

the scattering cross section 119862

119904120582 attenuation cross section

119862

119890120582 scattering factor 119876

119904120582 attenuation factor 119876

119890120582 and the

scattering phase function Φ

120582can be computed with Mie

scattering model [13] as

119862

119904120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11a)

119862

119890120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11b)

119876

119904120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11c)

119876

119890120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11d)

Φ

120582(120579) =

2

119876

119904120582120594

2[

1003816

1003816

1003816

1003816

119878

1

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119878

2

1003816

1003816

1003816

1003816

2

] (11e)

where 119886

119899and 119887

119899are Mie scattering coefficients and 119878

1

and 119878

2are amplitude functions Computation of the above

0

1

2

3

4

5

6

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

All

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

Figure 6 Calculated absorption coefficient of Al2O3particles with

Mie model

coefficients and functions has been illustrated in detail in [13]which is not repeated in this work

As for the spectral properties of the particle clouds thesingle and independent effect of each particle is assumedand all the spectral properties of the particle clouds can becomputed as the mathematic sum of all particles

The calculated absorption coefficients of 6 groups ofAl2O3particles with different diameters and the total value

of particles cloud in the center of plumersquos front end havebeen shown in Figure 6 That of the other two groupswith 119889 = 4 120583m 12 120583m is not included in the figure as theirconcentration is zero at front end of the plume One cansee that absorption coefficient of Al

2O3particles does not

change significantly with wavelength like that of plume gasesThe three groups of Al

2O3particles with smaller diameters

(119889 = 4 120583m 6 120583m 8 120583m) have larger absorption coefficientCompared with Figure 5 it is found that the absorptioncoefficient of particles cloud ismuch larger than that of plumegases so Al

2O3particles are the major radiation composition

in the rocket plumeThe calculated scattering coefficients of the 6 groups of

Al2O3particles with different diameters and total value of

particles cloud in the center of plumersquos front face have beenshown in Figure 7 One can find that scattering coefficientfor Al

2O3particles with 119889 = 4 120583m is most strong which

is medium for particles with 119889 = 6 120583m and is very smallfor extra particles Compared with Figure 6 it is found thatscattering coefficient of particles cloud is much bigger thanthe absorption coefficient and the scattering coefficient willgrow up with the increasing wavelength

The calculated scattering phase functions of Al2O3parti-

cles cloud for different wavelengths in the center of plumersquosfront end have been shown in Figure 8The scattering angle 120579

in the figure means the angle between the incident radianceand scattering direction It can be seen that the scatteringphase function in forward directions is much larger than

Journal of Applied Mathematics 7

0

2

4

6

8

10

12

2 3 4 5 6

All

120582 (120583m)

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

120590(c

mminus

1 )

Figure 7 Calculated scattering coefficient of Al2O3particles with

Mie model

that in rear directions and scattering phase function instraight forward direction is the largest among all the fivewavelengths Comparing the results for the five wavelengthsin straight forward direction the smaller wavelength has alarger forward scattering phase function

7 Results and Discussion

The calculation method of spectral property of plume gasesand Al

2O3particles and infrared radiation intensity has been

verified with a reduced scaling solid rocket in a vacuumchamber Infrared radiation intensity in wavebands 27sim295 120583m and 42sim445 120583m at three zenith angles 120579 = 60∘90∘ and 120∘ have been calculated and compared with testresults The maximum difference between calculated and testresults is 209 which arrives at 120579 = 90∘ and the minimumdifference is 104 which arrives at 120579 = 60∘ [22]

Studying the similarities of plumersquos radiation intensityfrom solid rocket with different scale and ratio of plumeradiation between the reduced scaling and full size rockets inwavelengths 120582 = 2 120583m 3 120583m 4 120583m and 5 120583m and waveband2sim6120583m have been calculated and illustrated in Figure 9 Thehorizontal ordinate 119903 in the figure means the scaling ratioof rocket model and the vertical ordinate 119868119868

0means the

ratio of plumersquos radiation intensityThe three dashed lines arecorresponding to the 15 2 and 25 power function of scalingratio Results of 120579 = 0∘ 40∘ 80∘ 120∘ and 160∘ with 120593 = 30∘have been investigated It can be seen in the figure that ratioof radiation intensity increases with the increasing scalingratio for all the four wavelengths and waveband 2sim6 120583mTheincreasing rule of radiation intensity ratio for 120579 = 0∘ gets closeto the curve of 15 power mostly which gets closer to the 2-power function of scaling ratio for 120579 = 40∘ and comes up toreach 25 power of scaling ratio for 120579 = 120∘ and 160∘ This iscaused by changing the length of high radiance section and

01

1

10

100

0 30 60 90 120 150 180

Phas

e fun

ctio

nΦ

Scattering angle 120579

120582 = 2120583m120582 = 3120583m120582 = 4120583m

120582 = 5120583m120582 = 6120583m

Figure 8 Calculated scattering phase function of Al2O3particles

with Mie model

radiance attenuation of the cold gases in tail section of theplume As plumersquos radiance mainly comes from the centralstripe which has a higher temperature and concentration forboth gases and Al

2O3particles Radiance attenuation of the

tail cold gases increases more rapidly than the projected areaon the surface of the plume with the growth of the scalingratio making growing rate of 119868119868

0being smaller than that of

surface area the latter has a growing rate of 1199032 in small zenithangle With the increase of zenith angle the effect of radianceattenuation decreases greatly and causes the ratio of radiationintensity increasing to reach and surpass 119903

2In the viewing of spectral characteristics for solid rocket

plume with different scaling ratios the variation of radiationintensity with wavelength for the 9 reduced scaling rocketmodels and the full scaled one in two directions 120579 =30∘ and 90∘ with 120593 = 30∘ has been plotted in Figure 10One can see good similarity between spectral variations ofdifferent scaling rocket models There are two peak spectraof radiation intensity 27sim30 120583mand 43sim46 120583m Comparedwith Figure 5 it is found that the former spectrum is thefeature radiative spectrum of H

2O the latter is of CO

2and

CO Besides in the two peak spectra spectral radiationintensity for plume of the 10 rockets decreases with theincreasing wavelength in 2sim6 120583mwaveband Since the Al

2O3

particle is the major radiation composition in the plumewhich has the maximum radiation for 120582 lt 20 120583m as particlethe temperature is higher than 2500K

The spectral radiation intensity of plume in all the zenithangles with 120593 = 30∘ for the 05 scaling rocket model and thefull scaled one has been shown in Figure 11 results of wave-length 120582 = 2 120583m 3 120583m and 4 120583m have been plotted With theincrease of the zenith angle radiation intensity of the plumefirstly increases arrives at a maximum value in a middleangle and decreases after that angle since the projected areaon plume surface changes with the zenith angle Comparingradiation intensity of plume in the front and rear hemisphereitrsquos found that is stronger in the rear hemisphere As read endsurface of plume has a strong radiance in the rear hemisphere

8 Journal of Applied Mathematics

0

03

06

09

12

15

0 02 04 06 08 1

II 0

120579 = 0∘ 120593 = 30∘

(a)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 40∘ 120593 = 30∘

(b)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 80∘ 120593 = 30∘

(c)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 120∘ 120593 = 30∘

(d)

0

02

04

06

08

1

0 02 04 06 08 1

120582 = 2120583m120582 = 3120583m120582 = 4120583m120582 = 5120583m

120582 = 2sim6120583mr15

r2

r25

II 0

120579 = 160∘ 120593 = 30∘

(e)

Figure 9 Ratio of plume radiation in the reduced scaling and full size rockets

Journal of Applied Mathematics 9

0

10000

20000

30000

40000

2 3 4 5 6

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

120582 (120583m)

I 120582W

(srmiddot120583

m)

(a)

0

10000

20000

30000

40000

I 120582W

(srmiddot120583

m)

2 3 4 5 6120582 (120583m)

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

(b)

Figure 10 Spectral radiation intensity of plume surface

0

2000

4000

6000

8000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

I 120582W

(srmiddot120583

m)

r = 05

(a)

0

10000

20000

30000

40000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

r = 1

I 120582W

(srmiddot120583

m)

(b)

Figure 11 Variation of radiation intensity of plume with zenith angles

Effect of radiance from the rear end also makes the directionwith maximum radiation intensity deflects toward the rearhemisphere which is remarkable for 119903 = 05 and themaximum radiation intensity arrives at 120579 = 50∘ in that caseThe colder section in the tail of plume is shorter and theradiance from the rear end surface would be bigger for asmaller scaling rocket

Figure 12 shows map of spectral radiance on 119909-119911 coordi-nate plane in plume flowof the 05 scaling solid rocket and thefull scaled ones for wavelength 120582 = 3 120583m and 5 120583m in 120579 = 0∘45∘ and 90∘ The radiance map of the two solid rockets hassimilar shape and close magnitude It is also found that the

high light area with a strong radiance is located in the centralstripe of the plume where temperature and concentrationof gases and Al

2O3particles are very high Compared the

width of the high light stripe in radiance maps of the twowavelengths it is narrower for 120582 = 3 120583m than 120582 = 5 120583mAs the radiation of Al

2O3particles plays the main role in

plume radiance of 3120583m the area with dense concentration ofparticles is smaller in radius than plume gases causing highradiance area centralizing in the plume in that wavelengthPlume gases are the major radiation composition for 120582 =5 120583m which has a bigger expansion angle and radius sizeand cause a wider stripe with high radiance Viewing the axial

10 Journal of Applied Mathematics

times102

z z z

z z z

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

r = 1 120582 = 3120583m 120579 = 0∘ r = 1 120582 = 3120583m 120579 = 45∘ r = 1 120582 = 3120583m 120579 = 90∘

r = 1 120582 = 5120583m 120579 = 0∘ r = 1 120582 = 5120583m 120579 = 45∘ r = 1 120582 = 5120583m 120579 = 90∘

1 2 3 4 5 6 7 8 9 10 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900times102

z

x

2 3 4 5 6 7 8 9 10 11 12

0

0 50 100 150 200 250 300

0 50 100 150 200 250 300 0 50 100 150 200 250 300

50 100 150 200 250 300 0

50

25

75

100

125

225

175

150

200

250

0 50 100 150 200 250

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

r = 05 120582 = 3120583m 120579 = 0∘ r = 05 120582 = 3120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

r = 05 120582 = 5120583m 120579 = 0∘ r = 05 120582 = 5120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

Figure 12 Map of spectral radiance on 119909-119911 coordinate plane

length of the high light stripe it is in the front part with halflength of the plume for the 05 scaling model but extendsfrom the front to end part of plume for the full scaled rocketmodel

8 Conclusion

Infrared radiation of plume from one full scaled solid rocketas well as the 01sim09 reducing scaled models in groundcondition has been investigated to study the similarity ofplumersquos infrared radiation for rocket models with differentscaling ratios Flowing field in the rocket and plume hasbeen computed with CFD code the omnidirectional andinfrared radiance on symmetric plane in the plume has beencalculated with the developed FVM codeThe research showsthat Al

2O3particle is the major radiation composition in the

rocket plumewhose scattering coefficient ismuch larger than

its absorption coefficient Ratio of radiation intensity fromreduced scaling plume to that of the full scaled one increaseswith the scaling ratio The power of the growth curve is 15for azimuth angle 120579 = 0∘ becomes 2 for 120579 = 40∘ and arrives at25 for 120579 = 120∘ and 160∘ There is good similarity betweenspectral variations of plumes from different scaling solidrockets and there are two peak spectra of radiation intensityin 27sim30 120583m and 43sim46 120583m Radiation intensity of theplume in rear hemisphere is bigger than that in the front andthe effect of radiance from the rear end surface makes thedirection with maximum radiation intensity deflects towardthe rear hemisphere

Conflict of Interests

There is not conflict of interests for all authors of this paper

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

temperatures and diameters The plume gases consist ofseveral polarity gases H

2O CO

2 CO HCL and OH whose

infrared radiative spectrum is made up with ten thousandsof lines [7] The common detailed calculating method for gasradiation uses a band model [8 9] or the weighted-sum-of-gray-gas (WSGG) model [10] The band model has a higherprecision for nonhomogeneous gases but is somewhat timelasting for large volume gases and is not very suitable forsolid rocket plume gases The WSGG model always showsa compromise of precision and calculation time which hasbeen widely used to calculate radiation of clouds or combus-tion gases As for spectral property of Al

2O3particles effects

of particle temperature concentrations and diameters areneeded to be considered [11 12] and theMie scattering theoryfor calculating particle radiation can also be used for Al

2O3

particles [13] Many numerical solution methods have beendeveloped to calculate radiation in nonhomogenous absorp-tiveemissivescatteringmediumby solving RTEnumericallyincluding Monte Carlo (MC) methods [14] the streamingmethod (SM) [15] discrete-ordinate method (DOM) [16]and finite volume methods (FVM) [17] To improve thesolution accuracy with less execution time the FVM hasoften been used to calculate the spectral radiation of thehigh temperature two-phase plume with highly expandingvolume

To study the similarity of infrared radiation from plumeswith different scaling ratio the Trident D5 solid rocket anda series of reduced scaling models with similar geometryhave been taken for investigation CFD code has been usedto compute the axial symmetric flowing field inside therocket engine and in plume of the full scaled rocket andreduced scaling model with same combustion chamberpressure and plumersquos flowing Mach number The spectralabsorption coefficient of plume gases H

2O CO

2 CO HCL

and OH will be computed with WSGG model Eight groupsof Al2O3particles with different diameters between 2 and

16 120583mhave been taken into considerationTheMie scatteringmodel has been used to compute the spectral absorptionscattering coefficient and phase function of Al

2O3particles

To calculate the spectral radiance of the plume a finitevolume method has been used to solve the RTE Integratingspectral radiance on exterior surface of plume volume in onedirection will give the plume radiation intensity Ratio ofradiation intensity between reduced scaling model and fullscaled one has been computed to study the similarity rules ofplume radiation

2 Calculation of Plume Flowing Field

The Trident D5 solid rocket used a composite propellantNEPE there is 10 weight of Al in the composite propellantPlumersquos flowing data of the full scaled rocket and reducedscalingmodels are calculated with the same input data for therocket nozzle pressure is 9Mpa and temperature is 3750KInput concentrations of major species are 119891OH = 0192 119891H

2

=0299 119891CO = 0027 119891CO

2

= 0146 and 119891H2O = 00008 on the

inlet of nozzle The full scaled rocket has a nozzle with 155mlength 035m diameter and 97 ratio of area expansionThe reduced scaling model has the same geometry but with

a minimized diameter and length To obtain the flowing dataof plume the CEA program has been adopted to compute thechemical-equilibrium compositions of the propellants andthe inlet parameters

Then CFD code is used to compute the united flowingin rocket nozzle and plume The time-marching method andthe advection upstream splitting method (AUSM) have beenused for numerical discretization of the 2D axis symmetricN-S equationsThe 119896-120576 turbulence model is adopted to simulateblending of the plume and atmosphere A finite rate chemicalreaction model with 12 components and 17 reactions hasbeen taken for consideration the detailed reaction equationsand related coefficients can be seen in [18] For the Al

2O3

particles the Lagrangian particle trajectory model has beenused to simulate exchange of energy andmomentumbetweenparticles and plume gases The distribution of the particlediameter is determined by Braithwaitersquos proposed functions[19]

The plumersquos flowing data of the full scaled Trident D5rocket and the 9 reduced scalingmodels have been calculatedin our study Only two groups of result are given in Figure 1One is for the full scaled rocket and the other is for the 05scaling model Considering the axial symmetric character-istics of the plumersquos flowing field only flowing field map onthe upper 119909-119911 plane has been plotted Figure 1 shows pressureand temperature of plume gases and volume fractions of the 3major components H

2O CO

2 and CO Volume fractions of

HCL and OH molecule have not been shown here as theirfractions are very small less than 005 Temperature andnumber concentration of the 8 groups of Al

2O3particles have

been calculated in our study only results of 3 groups withdiameters being 6 120583m 8 120583m and 10 120583m are given here

One can see from the figure that volume of the plumeexpands rapidly after the rocket nozzle making the diameterof the high temperature core at 119909 = 20m about 5 times that ofthe nozzle exit All maps of the flowing map of the 05 scalingrocket model have close similarity with the full scaled rocketwith similar parameter distributions and a close data rangeOne can also find that the high temperature areas of gases andAl2O3particles are a continuous central stripe in the plume

but the dense concentration area of Al2O3particles number

is quite discrete Comparing volume fraction of CO2and CO

it can be found that the high volume fraction area of CO2is

in the radial medium ring at some axial distance after thenozzle exit while that of CO is in the central thin stripe closebehind the nozzle exit For H

2O the high volume fraction

area is continuous in the central stripe of the plume

3 Calculation Method of Plume Radiation

A widely used equation of radiation transport in a nonho-mogeneous absorptiveemissivescattering medium can bewritten as variation of radiance 119894

1015840

120582(

S)while passing a distance119889S along the path S in the medium [20 21]

119889i1015840120582(S)

119889S= minus (Σ120581

119899+ 120590) i1015840120582(S) + (Σ120581

119899) i1015840120582119887

+

120590

4120587

int

4120587

i1015840120582(S1015840)Φ (S S1015840) 119889120596

1015840

(1)

Journal of Applied Mathematics 3

fCO2

1 32 4 6 85 7 9

2 4 6 8 10 12 1614 18

fH2O

0 84 12 2016 2824

tg

420

5 10 15

z

x

005 01 015 02 025

fCO

0

2E+10

3E+10

1E+10

4E+10

5E+10

6E+10

7E+10

500 1000 1500 2000 2500

tp3

0

12E

+10

6E+09

24E+10

18E+10

36E+10

3E+10

48E+10

42E+10

0

4E+9

6E+9

2E+9

8E+9

12E+10

14E+10

1E+10

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

prg2 4 6 8 10 12 14 16

times104times102 times10minus2

times10minus2

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

am3

am4 am5

(a) Plume flowing data of 05 scaling sized model

2 4 6 8 10 12 1614 18

fH2O

005 01 015 02 025

fCO

1E+10

2E+10

4E+10

3E+10

5E+10

0

1E+10

2E+10

4E+10

3E+10

5E+10

x

prg2 4 6 8 10 12 14 16

times104

tgtimes102

times10minus2

fCO2

1 32 4 6 85 7 9times10minus2

z

x

am3

500 1000 1500 2000 2500

tp3

0

4E+9

8E+9

12E+10

16E+10

2E+10

am4am5

2

1

02 4 6 8

z

2

1

02 4 6 8

z

x

2

1

02 4 6 8

x

z

x

2

1

02 4 6 8

z

2

1

02 4 6 8

z

x

2

1

02 4 6 8

x

z

x

2

1

02 4 6 8

z

2

1

02 4 6 8

z

x

2

1

02 4 6 8

0 4 8 12 16 24 28

(b) Plume flowing data of full scaled rocket

Figure 1 Plume flowing data of the 05 scaling model and full scaled solid rocket

4 Journal of Applied Mathematics

2

0

z(m

)

5 10 15 20

x (m)

Figure 2 Grids for radiance calculation in the plume

where the subscript ldquo120582rdquo means spectral variable and thesuperscript ldquo1015840rdquo means directional variable The symbol Σ120581

119899

is the summed absorption coefficient of plume gases andAl2O3particles and 120590 is the summed scattering coefficient

of 8 groups of Al2O3particles which are simply summed

up mathematically The symbol S1015840 means direction of theincident radiation 1205961015840 means the solid angle in S1015840 directionand Φ(S S1015840) means the scattering phase function betweenS and S1015840 The above symbols 120581

119899 120590 and Φ are all spectral

variables while the notations ldquo120582rdquo are omitted in (1) forabbreviation

According to finite volume method (1) can be solvedto compute spectral radiance of plume by integrating theequation on a control volume Considering the plume geom-etry the cylindrical control volumes had been firstly usedfor solving (1) but it is found that numerical convergence ofthe developed coefficient matrix is very poor being as thearea difference between the two radial opposite faces of thecontrol volume For that reason the cuboid control volumehas finally been chosen in our study The new computationdomain for radiation computation has the same length withthe flowing domain in Figure 1 but it has a square formend with side length being equal to the outer radius of theplume The precision of plume radiance solution is mainlyaffected by flowing data temperature volume fraction ornumber concentration of plume gases and Al

2O3particles

Those flowing data do not change dramatically in the plumeas seen in Figure 1 so the mesh size for radiance solution canbe much larger than that used in CFD solution For plume ofthe full scaled solid rocket the meshing plot on lengthwisesection of the computing domain is shown in Figure 2 Thedivided grids number is 80 times 20 in the upper 119909-119911 planeFlowing data of control volume inside the plume takes thearithmetic average of data on CFD grids in the volume Thatof control volume outside the plume is set to zero In thatcase all control volumes outside the plumewould not actuallyinfluence the result of plume radiance but the integral RTEequation is uniform for all the control volumes and solutionconvergence can be improved

Spectral radiance of each control volume is consideredin a series of directions every direction is correspondingto a set of angle zenith angle and azimuth angle Thezenith angle 120579 is defined as the angle between the plumecentral line (119909-axis) and the direction vector and azimuthangle 120593 is defined as angle between the projected directionvectors on 119909-119911 coordinate plane and the 119911 axis For radianceon control volume 119875 in direction S1015840 integrating (1) withGaussian integral method radiance of 119875 can be related toradiance of its neighbored control volumes in that directionThe detailed derivation procedure can be found in [19] onlythe final integration equation is given here as follows

f

w

NF

R

S

EW P

rs

n

x

y

e

z

Figure 3 A control volume and the 6 neighbors

max [Δ119860

119908119863

119908 0] i1015840119875+ max [Δ119860

119890119863

119890 0] i1015840119875

+ max [Δ119860

119904119863

119904 0] i1015840119875+ max [Δ119860

119899119863

119899 0] i1015840119875

+ max [Δ119860

119903119863

119903 0] i1015840119875+ max [Δ119860

119891119863

119891 0] i1015840119875

+ (Σ120581

119899+ 120590)Δ119881

119875Δ120596

1015840i1015840119875

= max [minusΔ119860

119908119863

119908 0] i1015840119882

+ max [minusΔ119860

119890119863

119890 0] i1015840119864

+ max [minusΔ119860

119904119863

119904 0] i1015840119878+ max [minusΔ119860

119899119863

119899 0] i1015840119873

+ max [minusΔ119860

119903119863

119903 0] i1015840119877+ max [minusΔ119860

119891119863

119891 0] i1015840119865

+ [Σ (120581

119899) i1015840120582119887

+

120590

4120587

Σ (i1015840119875(S) Φ (S S1015840) Δ120596)]Δ119881

119875Δ120596

1015840

119863

119896= int

Δ1205961015840

(S sdot n119896) 119889120596

1015840 119896 = 119908 119890 119899 119904 119903 119891

Φ (S S1015840) =

int

Δ1205961015840int

Δ120596Φ(S S1015840) 119889120596

1015840119889120596

Δ120596

1015840Δ120596

Δ120596

1015840= int

120593119897+

120593119897minus

int

120579119897+

120579119897minus

sin 120579 119889120579 119889120593

(2)

where subscripts 119882 119864 119873 119878 119877 and 119865 mean the variables ofthe neighbored control volume and 119908 119890 119899 119904 119903 and 119891 meanthe variables on interface of neighboring volumes as shownin Figure 3 The symbol Δ119860

119896is the area of the interface n

119896is

external normal vector of interface and Δ119881

119875is the volume of

control volumeThe upper equations can be also written as the following

simplified form

119886

119875i1015840119875

= 119886

119882i1015840119882

+ 119886

119864i1015840119864+ 119886

119878i1015840119878+ 119886

119873i1015840119873

+ 119886

119877i1015840119877+ 119886

119865i1015840119865+ b119875

119886

119875= Σmax (Δ119860

119896119863

119896 0)

Journal of Applied Mathematics 5

+ (Σ120581

119899+ 120590 minus

120590

4120587

Φ (S1015840 S1015840) Δ120596

1015840)Δ119881

119875Δ120596

1015840

b119875

= [(Σ120581

119899) i1015840120582119887

+

120590

4120587

sum

S =S1015840i1015840119875(S) Φ (S S1015840) Δ120596]Δ119881

119875Δ120596

1015840

(3)

The compound integral RTE equations of all control volumeshave the form of vector equations

[A] [I] = [B] (4)

Equation (4) is a nonlinear system of equations as variableb119875in the right side is actually unknown and needs to be

determined by radiance of control volume 119875 in all the otherdirections S1015840 So the solution of (4) could not be completedwith any iteration algorithm in one loop A cyclic iterationalgorithm with modification of b

119875has been proposed in this

study In each cycle b119875is firstly calculated with the radiance

computed from the last cycle Then a Gauss iteration algo-rithm will be used to solve vector equations (4) to computea new group of radiance for each control volume The cycliciteration will go on till the following convergence limitationis met

max i119899119875minus i119899minus1119875

le 120576 (5)

Considering the symmetric characteristics of geometry andflowing data of the plume radiance of plume is also sym-metric So we do not need to solve the vector equations ofRTE on all control volumes in the plume but only thosecontrol volumes with centers on 119909-119911 coordinate plane havebeen chosen for calculation While radiance of neighboredcontrol volumes is also required in (4) those control volumesare not centering on the 119909-119911 coordinate plane and a radiancetransformation algorithm on symmetric control volumes hasbeen proposed in our study to calculate the radiance of othercontrol volumes in the plume The radiance transformationalgorithm is derived based on rotation rules of vectors seenin Figure 4

If the radiance of control volume 119875 in direction S1is

i1015840119875(S1) the radiance of control volume 119874 in direction S

2is

equal to i1015840119875(S1)when the following vector equation is satisfied

p2= S2+ r2 (6)

4 Calculating Radiation Intensity fromthe Plume Surface

In the cases of studying base heating or heat protection of thesolid rocket spectral radiation intensity on the whole surfaceof the plume is concerned Spectral radiation intensity is alsoa directional variable which is the integration of spectralradiance i1015840

120582(S) on surface of bounding control volumes

in direction S If radiation intensity in some waveband

z

PQ

y

1198511

1198261

1198262

1198262

1198491

11984921198512

120575120575

120579

120579

Figure 4 Radiance transformation on symmetric control volumes

is concerned as for the 2sim6120583m band integration on thewavelength is also needed

119868

120582= sum

119895119860

i1015840120582(S) eS sdot n

Δ119860Δ119860 eS sdot n

Δ119860gt 0

119868 = sum

119895120582

sum

119895119860

i1015840 (S) eS sdot nΔ119860

Δ119860Δ120582

(7)

where 119895

119860is number of control volume surfaces on boundary

of the plume eS is unit vector of the radiance direction andnΔ119860

is unit vector of external normal on the surface

5 Calculation of Spectral Property ofPlume Gases and Al2O3 Particles

The3major radiation gases of the solid rocket plume areH2O

CO2 and CO To calculate the spectral absorption coefficient

of each plume gas the standard absorption coefficient 120581120582STP

at several temperatures and 1 atm pressure in [20 21] has beenused For plume gas with pressure being 119875 and temperaturebeing 119879 absorption coefficient can be modified from 120581

120582STPas follows

120581

120582=

119875

10

5

273

119879

120581

120582STP (8)

Absorption coefficient of the plume gases mixture is calcu-lated with the WSGG model and the weight is the volumefraction of each gas If the volume fraction of one gas is 119891then the absorption coefficient of the gases mixture can bewritten with the WSGG model as

120581

120582119892= 120581

120582H2O119891H

2O + 120581

120582CO2

119891CO2

+ 120581

120582CO119891CO (9)

The calculated absorption coefficient for plume gases H2O

CO2 and CO and the gases mixture in center of front end of

the plume have been shown in Figure 5 One can find thatabsorption coefficient of CO

2and CO is comparatively much

larger than that of CO is 42sim55 120583m the peak spectrum ofCO2is 42sim5 120583m and absorption coefficient of H

2O is much

smaller Absorption coefficient of the gases mixture appearsas an accumulated effect of the three gases

6 Journal of Applied Mathematics

0

1

2

3

4

5

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

kH2OkCO2kCO

kHCLkgases

Figure 5 Calculated absorption coefficient of plume gases withWSGG model

6 Calculation of the Spectral Property ofAl2O3 Particles

In the calculation of spectral properties of Al2O3particles

with Mie scattering model particle size number concen-tration and complex refractive index must be determinedfirstly Asmentioned above size andnumber concentration ofAl2O3particles have been calculated in plume flowing com-

putation The complex refractive index of Al2O3particles

m = 119899 + i119896 takes formulas recommended by Reed et al [12]

119896 = 466 times 10

minus4120582

133119879

15 exp [minus

29420

119879

]

119899 = 175 cos (6120582) (10)

For a single Al2O3particle spectral properties including

the scattering cross section 119862

119904120582 attenuation cross section

119862

119890120582 scattering factor 119876

119904120582 attenuation factor 119876

119890120582 and the

scattering phase function Φ

120582can be computed with Mie

scattering model [13] as

119862

119904120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11a)

119862

119890120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11b)

119876

119904120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11c)

119876

119890120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11d)

Φ

120582(120579) =

2

119876

119904120582120594

2[

1003816

1003816

1003816

1003816

119878

1

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119878

2

1003816

1003816

1003816

1003816

2

] (11e)

where 119886

119899and 119887

119899are Mie scattering coefficients and 119878

1

and 119878

2are amplitude functions Computation of the above

0

1

2

3

4

5

6

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

All

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

Figure 6 Calculated absorption coefficient of Al2O3particles with

Mie model

coefficients and functions has been illustrated in detail in [13]which is not repeated in this work

As for the spectral properties of the particle clouds thesingle and independent effect of each particle is assumedand all the spectral properties of the particle clouds can becomputed as the mathematic sum of all particles

The calculated absorption coefficients of 6 groups ofAl2O3particles with different diameters and the total value

of particles cloud in the center of plumersquos front end havebeen shown in Figure 6 That of the other two groupswith 119889 = 4 120583m 12 120583m is not included in the figure as theirconcentration is zero at front end of the plume One cansee that absorption coefficient of Al

2O3particles does not

change significantly with wavelength like that of plume gasesThe three groups of Al

2O3particles with smaller diameters

(119889 = 4 120583m 6 120583m 8 120583m) have larger absorption coefficientCompared with Figure 5 it is found that the absorptioncoefficient of particles cloud ismuch larger than that of plumegases so Al

2O3particles are the major radiation composition

in the rocket plumeThe calculated scattering coefficients of the 6 groups of

Al2O3particles with different diameters and total value of

particles cloud in the center of plumersquos front face have beenshown in Figure 7 One can find that scattering coefficientfor Al

2O3particles with 119889 = 4 120583m is most strong which

is medium for particles with 119889 = 6 120583m and is very smallfor extra particles Compared with Figure 6 it is found thatscattering coefficient of particles cloud is much bigger thanthe absorption coefficient and the scattering coefficient willgrow up with the increasing wavelength

The calculated scattering phase functions of Al2O3parti-

cles cloud for different wavelengths in the center of plumersquosfront end have been shown in Figure 8The scattering angle 120579

in the figure means the angle between the incident radianceand scattering direction It can be seen that the scatteringphase function in forward directions is much larger than

Journal of Applied Mathematics 7

0

2

4

6

8

10

12

2 3 4 5 6

All

120582 (120583m)

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

120590(c

mminus

1 )

Figure 7 Calculated scattering coefficient of Al2O3particles with

Mie model

that in rear directions and scattering phase function instraight forward direction is the largest among all the fivewavelengths Comparing the results for the five wavelengthsin straight forward direction the smaller wavelength has alarger forward scattering phase function

7 Results and Discussion

The calculation method of spectral property of plume gasesand Al

2O3particles and infrared radiation intensity has been

verified with a reduced scaling solid rocket in a vacuumchamber Infrared radiation intensity in wavebands 27sim295 120583m and 42sim445 120583m at three zenith angles 120579 = 60∘90∘ and 120∘ have been calculated and compared with testresults The maximum difference between calculated and testresults is 209 which arrives at 120579 = 90∘ and the minimumdifference is 104 which arrives at 120579 = 60∘ [22]

Studying the similarities of plumersquos radiation intensityfrom solid rocket with different scale and ratio of plumeradiation between the reduced scaling and full size rockets inwavelengths 120582 = 2 120583m 3 120583m 4 120583m and 5 120583m and waveband2sim6120583m have been calculated and illustrated in Figure 9 Thehorizontal ordinate 119903 in the figure means the scaling ratioof rocket model and the vertical ordinate 119868119868

0means the

ratio of plumersquos radiation intensityThe three dashed lines arecorresponding to the 15 2 and 25 power function of scalingratio Results of 120579 = 0∘ 40∘ 80∘ 120∘ and 160∘ with 120593 = 30∘have been investigated It can be seen in the figure that ratioof radiation intensity increases with the increasing scalingratio for all the four wavelengths and waveband 2sim6 120583mTheincreasing rule of radiation intensity ratio for 120579 = 0∘ gets closeto the curve of 15 power mostly which gets closer to the 2-power function of scaling ratio for 120579 = 40∘ and comes up toreach 25 power of scaling ratio for 120579 = 120∘ and 160∘ This iscaused by changing the length of high radiance section and

01

1

10

100

0 30 60 90 120 150 180

Phas

e fun

ctio

nΦ

Scattering angle 120579

120582 = 2120583m120582 = 3120583m120582 = 4120583m

120582 = 5120583m120582 = 6120583m

Figure 8 Calculated scattering phase function of Al2O3particles

with Mie model

radiance attenuation of the cold gases in tail section of theplume As plumersquos radiance mainly comes from the centralstripe which has a higher temperature and concentration forboth gases and Al

2O3particles Radiance attenuation of the

tail cold gases increases more rapidly than the projected areaon the surface of the plume with the growth of the scalingratio making growing rate of 119868119868

0being smaller than that of

surface area the latter has a growing rate of 1199032 in small zenithangle With the increase of zenith angle the effect of radianceattenuation decreases greatly and causes the ratio of radiationintensity increasing to reach and surpass 119903

2In the viewing of spectral characteristics for solid rocket

plume with different scaling ratios the variation of radiationintensity with wavelength for the 9 reduced scaling rocketmodels and the full scaled one in two directions 120579 =30∘ and 90∘ with 120593 = 30∘ has been plotted in Figure 10One can see good similarity between spectral variations ofdifferent scaling rocket models There are two peak spectraof radiation intensity 27sim30 120583mand 43sim46 120583m Comparedwith Figure 5 it is found that the former spectrum is thefeature radiative spectrum of H

2O the latter is of CO

2and

CO Besides in the two peak spectra spectral radiationintensity for plume of the 10 rockets decreases with theincreasing wavelength in 2sim6 120583mwaveband Since the Al

2O3

particle is the major radiation composition in the plumewhich has the maximum radiation for 120582 lt 20 120583m as particlethe temperature is higher than 2500K

The spectral radiation intensity of plume in all the zenithangles with 120593 = 30∘ for the 05 scaling rocket model and thefull scaled one has been shown in Figure 11 results of wave-length 120582 = 2 120583m 3 120583m and 4 120583m have been plotted With theincrease of the zenith angle radiation intensity of the plumefirstly increases arrives at a maximum value in a middleangle and decreases after that angle since the projected areaon plume surface changes with the zenith angle Comparingradiation intensity of plume in the front and rear hemisphereitrsquos found that is stronger in the rear hemisphere As read endsurface of plume has a strong radiance in the rear hemisphere

8 Journal of Applied Mathematics

0

03

06

09

12

15

0 02 04 06 08 1

II 0

120579 = 0∘ 120593 = 30∘

(a)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 40∘ 120593 = 30∘

(b)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 80∘ 120593 = 30∘

(c)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 120∘ 120593 = 30∘

(d)

0

02

04

06

08

1

0 02 04 06 08 1

120582 = 2120583m120582 = 3120583m120582 = 4120583m120582 = 5120583m

120582 = 2sim6120583mr15

r2

r25

II 0

120579 = 160∘ 120593 = 30∘

(e)

Figure 9 Ratio of plume radiation in the reduced scaling and full size rockets

Journal of Applied Mathematics 9

0

10000

20000

30000

40000

2 3 4 5 6

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

120582 (120583m)

I 120582W

(srmiddot120583

m)

(a)

0

10000

20000

30000

40000

I 120582W

(srmiddot120583

m)

2 3 4 5 6120582 (120583m)

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

(b)

Figure 10 Spectral radiation intensity of plume surface

0

2000

4000

6000

8000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

I 120582W

(srmiddot120583

m)

r = 05

(a)

0

10000

20000

30000

40000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

r = 1

I 120582W

(srmiddot120583

m)

(b)

Figure 11 Variation of radiation intensity of plume with zenith angles

Effect of radiance from the rear end also makes the directionwith maximum radiation intensity deflects toward the rearhemisphere which is remarkable for 119903 = 05 and themaximum radiation intensity arrives at 120579 = 50∘ in that caseThe colder section in the tail of plume is shorter and theradiance from the rear end surface would be bigger for asmaller scaling rocket

Figure 12 shows map of spectral radiance on 119909-119911 coordi-nate plane in plume flowof the 05 scaling solid rocket and thefull scaled ones for wavelength 120582 = 3 120583m and 5 120583m in 120579 = 0∘45∘ and 90∘ The radiance map of the two solid rockets hassimilar shape and close magnitude It is also found that the

high light area with a strong radiance is located in the centralstripe of the plume where temperature and concentrationof gases and Al

2O3particles are very high Compared the

width of the high light stripe in radiance maps of the twowavelengths it is narrower for 120582 = 3 120583m than 120582 = 5 120583mAs the radiation of Al

2O3particles plays the main role in

plume radiance of 3120583m the area with dense concentration ofparticles is smaller in radius than plume gases causing highradiance area centralizing in the plume in that wavelengthPlume gases are the major radiation composition for 120582 =5 120583m which has a bigger expansion angle and radius sizeand cause a wider stripe with high radiance Viewing the axial

10 Journal of Applied Mathematics

times102

z z z

z z z

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

r = 1 120582 = 3120583m 120579 = 0∘ r = 1 120582 = 3120583m 120579 = 45∘ r = 1 120582 = 3120583m 120579 = 90∘

r = 1 120582 = 5120583m 120579 = 0∘ r = 1 120582 = 5120583m 120579 = 45∘ r = 1 120582 = 5120583m 120579 = 90∘

1 2 3 4 5 6 7 8 9 10 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900times102

z

x

2 3 4 5 6 7 8 9 10 11 12

0

0 50 100 150 200 250 300

0 50 100 150 200 250 300 0 50 100 150 200 250 300

50 100 150 200 250 300 0

50

25

75

100

125

225

175

150

200

250

0 50 100 150 200 250

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

r = 05 120582 = 3120583m 120579 = 0∘ r = 05 120582 = 3120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

r = 05 120582 = 5120583m 120579 = 0∘ r = 05 120582 = 5120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

Figure 12 Map of spectral radiance on 119909-119911 coordinate plane

length of the high light stripe it is in the front part with halflength of the plume for the 05 scaling model but extendsfrom the front to end part of plume for the full scaled rocketmodel

8 Conclusion

Infrared radiation of plume from one full scaled solid rocketas well as the 01sim09 reducing scaled models in groundcondition has been investigated to study the similarity ofplumersquos infrared radiation for rocket models with differentscaling ratios Flowing field in the rocket and plume hasbeen computed with CFD code the omnidirectional andinfrared radiance on symmetric plane in the plume has beencalculated with the developed FVM codeThe research showsthat Al

2O3particle is the major radiation composition in the

rocket plumewhose scattering coefficient ismuch larger than

its absorption coefficient Ratio of radiation intensity fromreduced scaling plume to that of the full scaled one increaseswith the scaling ratio The power of the growth curve is 15for azimuth angle 120579 = 0∘ becomes 2 for 120579 = 40∘ and arrives at25 for 120579 = 120∘ and 160∘ There is good similarity betweenspectral variations of plumes from different scaling solidrockets and there are two peak spectra of radiation intensityin 27sim30 120583m and 43sim46 120583m Radiation intensity of theplume in rear hemisphere is bigger than that in the front andthe effect of radiance from the rear end surface makes thedirection with maximum radiation intensity deflects towardthe rear hemisphere

Conflict of Interests

There is not conflict of interests for all authors of this paper

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

fCO2

1 32 4 6 85 7 9

2 4 6 8 10 12 1614 18

fH2O

0 84 12 2016 2824

tg

420

5 10 15

z

x

005 01 015 02 025

fCO

0

2E+10

3E+10

1E+10

4E+10

5E+10

6E+10

7E+10

500 1000 1500 2000 2500

tp3

0

12E

+10

6E+09

24E+10

18E+10

36E+10

3E+10

48E+10

42E+10

0

4E+9

6E+9

2E+9

8E+9

12E+10

14E+10

1E+10

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

prg2 4 6 8 10 12 14 16

times104times102 times10minus2

times10minus2

420

5 10 15

z

x

420

5 10 15

z

x

420

5 10 15

z

x

am3

am4 am5

(a) Plume flowing data of 05 scaling sized model

2 4 6 8 10 12 1614 18

fH2O

005 01 015 02 025

fCO

1E+10

2E+10

4E+10

3E+10

5E+10

0

1E+10

2E+10

4E+10

3E+10

5E+10

x

prg2 4 6 8 10 12 14 16

times104

tgtimes102

times10minus2

fCO2

1 32 4 6 85 7 9times10minus2

z

x

am3

500 1000 1500 2000 2500

tp3

0

4E+9

8E+9

12E+10

16E+10

2E+10

am4am5

2

1

02 4 6 8

z

2

1

02 4 6 8

z

x

2

1

02 4 6 8

x

z

x

2

1

02 4 6 8

z

2

1

02 4 6 8

z

x

2

1

02 4 6 8

x

z

x

2

1

02 4 6 8

z

2

1

02 4 6 8

z

x

2

1

02 4 6 8

0 4 8 12 16 24 28

(b) Plume flowing data of full scaled rocket

Figure 1 Plume flowing data of the 05 scaling model and full scaled solid rocket

4 Journal of Applied Mathematics

2

0

z(m

)

5 10 15 20

x (m)

Figure 2 Grids for radiance calculation in the plume

where the subscript ldquo120582rdquo means spectral variable and thesuperscript ldquo1015840rdquo means directional variable The symbol Σ120581

119899

is the summed absorption coefficient of plume gases andAl2O3particles and 120590 is the summed scattering coefficient

of 8 groups of Al2O3particles which are simply summed

up mathematically The symbol S1015840 means direction of theincident radiation 1205961015840 means the solid angle in S1015840 directionand Φ(S S1015840) means the scattering phase function betweenS and S1015840 The above symbols 120581

119899 120590 and Φ are all spectral

variables while the notations ldquo120582rdquo are omitted in (1) forabbreviation

According to finite volume method (1) can be solvedto compute spectral radiance of plume by integrating theequation on a control volume Considering the plume geom-etry the cylindrical control volumes had been firstly usedfor solving (1) but it is found that numerical convergence ofthe developed coefficient matrix is very poor being as thearea difference between the two radial opposite faces of thecontrol volume For that reason the cuboid control volumehas finally been chosen in our study The new computationdomain for radiation computation has the same length withthe flowing domain in Figure 1 but it has a square formend with side length being equal to the outer radius of theplume The precision of plume radiance solution is mainlyaffected by flowing data temperature volume fraction ornumber concentration of plume gases and Al

2O3particles

Those flowing data do not change dramatically in the plumeas seen in Figure 1 so the mesh size for radiance solution canbe much larger than that used in CFD solution For plume ofthe full scaled solid rocket the meshing plot on lengthwisesection of the computing domain is shown in Figure 2 Thedivided grids number is 80 times 20 in the upper 119909-119911 planeFlowing data of control volume inside the plume takes thearithmetic average of data on CFD grids in the volume Thatof control volume outside the plume is set to zero In thatcase all control volumes outside the plumewould not actuallyinfluence the result of plume radiance but the integral RTEequation is uniform for all the control volumes and solutionconvergence can be improved

Spectral radiance of each control volume is consideredin a series of directions every direction is correspondingto a set of angle zenith angle and azimuth angle Thezenith angle 120579 is defined as the angle between the plumecentral line (119909-axis) and the direction vector and azimuthangle 120593 is defined as angle between the projected directionvectors on 119909-119911 coordinate plane and the 119911 axis For radianceon control volume 119875 in direction S1015840 integrating (1) withGaussian integral method radiance of 119875 can be related toradiance of its neighbored control volumes in that directionThe detailed derivation procedure can be found in [19] onlythe final integration equation is given here as follows

f

w

NF

R

S

EW P

rs

n

x

y

e

z

Figure 3 A control volume and the 6 neighbors

max [Δ119860

119908119863

119908 0] i1015840119875+ max [Δ119860

119890119863

119890 0] i1015840119875

+ max [Δ119860

119904119863

119904 0] i1015840119875+ max [Δ119860

119899119863

119899 0] i1015840119875

+ max [Δ119860

119903119863

119903 0] i1015840119875+ max [Δ119860

119891119863

119891 0] i1015840119875

+ (Σ120581

119899+ 120590)Δ119881

119875Δ120596

1015840i1015840119875

= max [minusΔ119860

119908119863

119908 0] i1015840119882

+ max [minusΔ119860

119890119863

119890 0] i1015840119864

+ max [minusΔ119860

119904119863

119904 0] i1015840119878+ max [minusΔ119860

119899119863

119899 0] i1015840119873

+ max [minusΔ119860

119903119863

119903 0] i1015840119877+ max [minusΔ119860

119891119863

119891 0] i1015840119865

+ [Σ (120581

119899) i1015840120582119887

+

120590

4120587

Σ (i1015840119875(S) Φ (S S1015840) Δ120596)]Δ119881

119875Δ120596

1015840

119863

119896= int

Δ1205961015840

(S sdot n119896) 119889120596

1015840 119896 = 119908 119890 119899 119904 119903 119891

Φ (S S1015840) =

int

Δ1205961015840int

Δ120596Φ(S S1015840) 119889120596

1015840119889120596

Δ120596

1015840Δ120596

Δ120596

1015840= int

120593119897+

120593119897minus

int

120579119897+

120579119897minus

sin 120579 119889120579 119889120593

(2)

where subscripts 119882 119864 119873 119878 119877 and 119865 mean the variables ofthe neighbored control volume and 119908 119890 119899 119904 119903 and 119891 meanthe variables on interface of neighboring volumes as shownin Figure 3 The symbol Δ119860

119896is the area of the interface n

119896is

external normal vector of interface and Δ119881

119875is the volume of

control volumeThe upper equations can be also written as the following

simplified form

119886

119875i1015840119875

= 119886

119882i1015840119882

+ 119886

119864i1015840119864+ 119886

119878i1015840119878+ 119886

119873i1015840119873

+ 119886

119877i1015840119877+ 119886

119865i1015840119865+ b119875

119886

119875= Σmax (Δ119860

119896119863

119896 0)

Journal of Applied Mathematics 5

+ (Σ120581

119899+ 120590 minus

120590

4120587

Φ (S1015840 S1015840) Δ120596

1015840)Δ119881

119875Δ120596

1015840

b119875

= [(Σ120581

119899) i1015840120582119887

+

120590

4120587

sum

S =S1015840i1015840119875(S) Φ (S S1015840) Δ120596]Δ119881

119875Δ120596

1015840

(3)

The compound integral RTE equations of all control volumeshave the form of vector equations

[A] [I] = [B] (4)

Equation (4) is a nonlinear system of equations as variableb119875in the right side is actually unknown and needs to be

determined by radiance of control volume 119875 in all the otherdirections S1015840 So the solution of (4) could not be completedwith any iteration algorithm in one loop A cyclic iterationalgorithm with modification of b

119875has been proposed in this

study In each cycle b119875is firstly calculated with the radiance

computed from the last cycle Then a Gauss iteration algo-rithm will be used to solve vector equations (4) to computea new group of radiance for each control volume The cycliciteration will go on till the following convergence limitationis met

max i119899119875minus i119899minus1119875

le 120576 (5)

Considering the symmetric characteristics of geometry andflowing data of the plume radiance of plume is also sym-metric So we do not need to solve the vector equations ofRTE on all control volumes in the plume but only thosecontrol volumes with centers on 119909-119911 coordinate plane havebeen chosen for calculation While radiance of neighboredcontrol volumes is also required in (4) those control volumesare not centering on the 119909-119911 coordinate plane and a radiancetransformation algorithm on symmetric control volumes hasbeen proposed in our study to calculate the radiance of othercontrol volumes in the plume The radiance transformationalgorithm is derived based on rotation rules of vectors seenin Figure 4

If the radiance of control volume 119875 in direction S1is

i1015840119875(S1) the radiance of control volume 119874 in direction S

2is

equal to i1015840119875(S1)when the following vector equation is satisfied

p2= S2+ r2 (6)

4 Calculating Radiation Intensity fromthe Plume Surface

In the cases of studying base heating or heat protection of thesolid rocket spectral radiation intensity on the whole surfaceof the plume is concerned Spectral radiation intensity is alsoa directional variable which is the integration of spectralradiance i1015840

120582(S) on surface of bounding control volumes

in direction S If radiation intensity in some waveband

z

PQ

y

1198511

1198261

1198262

1198262

1198491

11984921198512

120575120575

120579

120579

Figure 4 Radiance transformation on symmetric control volumes

is concerned as for the 2sim6120583m band integration on thewavelength is also needed

119868

120582= sum

119895119860

i1015840120582(S) eS sdot n

Δ119860Δ119860 eS sdot n

Δ119860gt 0

119868 = sum

119895120582

sum

119895119860

i1015840 (S) eS sdot nΔ119860

Δ119860Δ120582

(7)

where 119895

119860is number of control volume surfaces on boundary

of the plume eS is unit vector of the radiance direction andnΔ119860

is unit vector of external normal on the surface

5 Calculation of Spectral Property ofPlume Gases and Al2O3 Particles

The3major radiation gases of the solid rocket plume areH2O

CO2 and CO To calculate the spectral absorption coefficient

of each plume gas the standard absorption coefficient 120581120582STP

at several temperatures and 1 atm pressure in [20 21] has beenused For plume gas with pressure being 119875 and temperaturebeing 119879 absorption coefficient can be modified from 120581

120582STPas follows

120581

120582=

119875

10

5

273

119879

120581

120582STP (8)

Absorption coefficient of the plume gases mixture is calcu-lated with the WSGG model and the weight is the volumefraction of each gas If the volume fraction of one gas is 119891then the absorption coefficient of the gases mixture can bewritten with the WSGG model as

120581

120582119892= 120581

120582H2O119891H

2O + 120581

120582CO2

119891CO2

+ 120581

120582CO119891CO (9)

The calculated absorption coefficient for plume gases H2O

CO2 and CO and the gases mixture in center of front end of

the plume have been shown in Figure 5 One can find thatabsorption coefficient of CO

2and CO is comparatively much

larger than that of CO is 42sim55 120583m the peak spectrum ofCO2is 42sim5 120583m and absorption coefficient of H

2O is much

smaller Absorption coefficient of the gases mixture appearsas an accumulated effect of the three gases

6 Journal of Applied Mathematics

0

1

2

3

4

5

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

kH2OkCO2kCO

kHCLkgases

Figure 5 Calculated absorption coefficient of plume gases withWSGG model

6 Calculation of the Spectral Property ofAl2O3 Particles

In the calculation of spectral properties of Al2O3particles

with Mie scattering model particle size number concen-tration and complex refractive index must be determinedfirstly Asmentioned above size andnumber concentration ofAl2O3particles have been calculated in plume flowing com-

putation The complex refractive index of Al2O3particles

m = 119899 + i119896 takes formulas recommended by Reed et al [12]

119896 = 466 times 10

minus4120582

133119879

15 exp [minus

29420

119879

]

119899 = 175 cos (6120582) (10)

For a single Al2O3particle spectral properties including

the scattering cross section 119862

119904120582 attenuation cross section

119862

119890120582 scattering factor 119876

119904120582 attenuation factor 119876

119890120582 and the

scattering phase function Φ

120582can be computed with Mie

scattering model [13] as

119862

119904120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11a)

119862

119890120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11b)

119876

119904120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11c)

119876

119890120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11d)

Φ

120582(120579) =

2

119876

119904120582120594

2[

1003816

1003816

1003816

1003816

119878

1

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119878

2

1003816

1003816

1003816

1003816

2

] (11e)

where 119886

119899and 119887

119899are Mie scattering coefficients and 119878

1

and 119878

2are amplitude functions Computation of the above

0

1

2

3

4

5

6

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

All

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

Figure 6 Calculated absorption coefficient of Al2O3particles with

Mie model

coefficients and functions has been illustrated in detail in [13]which is not repeated in this work

As for the spectral properties of the particle clouds thesingle and independent effect of each particle is assumedand all the spectral properties of the particle clouds can becomputed as the mathematic sum of all particles

The calculated absorption coefficients of 6 groups ofAl2O3particles with different diameters and the total value

of particles cloud in the center of plumersquos front end havebeen shown in Figure 6 That of the other two groupswith 119889 = 4 120583m 12 120583m is not included in the figure as theirconcentration is zero at front end of the plume One cansee that absorption coefficient of Al

2O3particles does not

change significantly with wavelength like that of plume gasesThe three groups of Al

2O3particles with smaller diameters

(119889 = 4 120583m 6 120583m 8 120583m) have larger absorption coefficientCompared with Figure 5 it is found that the absorptioncoefficient of particles cloud ismuch larger than that of plumegases so Al

2O3particles are the major radiation composition

in the rocket plumeThe calculated scattering coefficients of the 6 groups of

Al2O3particles with different diameters and total value of

particles cloud in the center of plumersquos front face have beenshown in Figure 7 One can find that scattering coefficientfor Al

2O3particles with 119889 = 4 120583m is most strong which

is medium for particles with 119889 = 6 120583m and is very smallfor extra particles Compared with Figure 6 it is found thatscattering coefficient of particles cloud is much bigger thanthe absorption coefficient and the scattering coefficient willgrow up with the increasing wavelength

The calculated scattering phase functions of Al2O3parti-

cles cloud for different wavelengths in the center of plumersquosfront end have been shown in Figure 8The scattering angle 120579

in the figure means the angle between the incident radianceand scattering direction It can be seen that the scatteringphase function in forward directions is much larger than

Journal of Applied Mathematics 7

0

2

4

6

8

10

12

2 3 4 5 6

All

120582 (120583m)

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

120590(c

mminus

1 )

Figure 7 Calculated scattering coefficient of Al2O3particles with

Mie model

that in rear directions and scattering phase function instraight forward direction is the largest among all the fivewavelengths Comparing the results for the five wavelengthsin straight forward direction the smaller wavelength has alarger forward scattering phase function

7 Results and Discussion

The calculation method of spectral property of plume gasesand Al

2O3particles and infrared radiation intensity has been

verified with a reduced scaling solid rocket in a vacuumchamber Infrared radiation intensity in wavebands 27sim295 120583m and 42sim445 120583m at three zenith angles 120579 = 60∘90∘ and 120∘ have been calculated and compared with testresults The maximum difference between calculated and testresults is 209 which arrives at 120579 = 90∘ and the minimumdifference is 104 which arrives at 120579 = 60∘ [22]

Studying the similarities of plumersquos radiation intensityfrom solid rocket with different scale and ratio of plumeradiation between the reduced scaling and full size rockets inwavelengths 120582 = 2 120583m 3 120583m 4 120583m and 5 120583m and waveband2sim6120583m have been calculated and illustrated in Figure 9 Thehorizontal ordinate 119903 in the figure means the scaling ratioof rocket model and the vertical ordinate 119868119868

0means the

ratio of plumersquos radiation intensityThe three dashed lines arecorresponding to the 15 2 and 25 power function of scalingratio Results of 120579 = 0∘ 40∘ 80∘ 120∘ and 160∘ with 120593 = 30∘have been investigated It can be seen in the figure that ratioof radiation intensity increases with the increasing scalingratio for all the four wavelengths and waveband 2sim6 120583mTheincreasing rule of radiation intensity ratio for 120579 = 0∘ gets closeto the curve of 15 power mostly which gets closer to the 2-power function of scaling ratio for 120579 = 40∘ and comes up toreach 25 power of scaling ratio for 120579 = 120∘ and 160∘ This iscaused by changing the length of high radiance section and

01

1

10

100

0 30 60 90 120 150 180

Phas

e fun

ctio

nΦ

Scattering angle 120579

120582 = 2120583m120582 = 3120583m120582 = 4120583m

120582 = 5120583m120582 = 6120583m

Figure 8 Calculated scattering phase function of Al2O3particles

with Mie model

radiance attenuation of the cold gases in tail section of theplume As plumersquos radiance mainly comes from the centralstripe which has a higher temperature and concentration forboth gases and Al

2O3particles Radiance attenuation of the

tail cold gases increases more rapidly than the projected areaon the surface of the plume with the growth of the scalingratio making growing rate of 119868119868

0being smaller than that of

surface area the latter has a growing rate of 1199032 in small zenithangle With the increase of zenith angle the effect of radianceattenuation decreases greatly and causes the ratio of radiationintensity increasing to reach and surpass 119903

2In the viewing of spectral characteristics for solid rocket

plume with different scaling ratios the variation of radiationintensity with wavelength for the 9 reduced scaling rocketmodels and the full scaled one in two directions 120579 =30∘ and 90∘ with 120593 = 30∘ has been plotted in Figure 10One can see good similarity between spectral variations ofdifferent scaling rocket models There are two peak spectraof radiation intensity 27sim30 120583mand 43sim46 120583m Comparedwith Figure 5 it is found that the former spectrum is thefeature radiative spectrum of H

2O the latter is of CO

2and

CO Besides in the two peak spectra spectral radiationintensity for plume of the 10 rockets decreases with theincreasing wavelength in 2sim6 120583mwaveband Since the Al

2O3

particle is the major radiation composition in the plumewhich has the maximum radiation for 120582 lt 20 120583m as particlethe temperature is higher than 2500K

The spectral radiation intensity of plume in all the zenithangles with 120593 = 30∘ for the 05 scaling rocket model and thefull scaled one has been shown in Figure 11 results of wave-length 120582 = 2 120583m 3 120583m and 4 120583m have been plotted With theincrease of the zenith angle radiation intensity of the plumefirstly increases arrives at a maximum value in a middleangle and decreases after that angle since the projected areaon plume surface changes with the zenith angle Comparingradiation intensity of plume in the front and rear hemisphereitrsquos found that is stronger in the rear hemisphere As read endsurface of plume has a strong radiance in the rear hemisphere

8 Journal of Applied Mathematics

0

03

06

09

12

15

0 02 04 06 08 1

II 0

120579 = 0∘ 120593 = 30∘

(a)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 40∘ 120593 = 30∘

(b)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 80∘ 120593 = 30∘

(c)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 120∘ 120593 = 30∘

(d)

0

02

04

06

08

1

0 02 04 06 08 1

120582 = 2120583m120582 = 3120583m120582 = 4120583m120582 = 5120583m

120582 = 2sim6120583mr15

r2

r25

II 0

120579 = 160∘ 120593 = 30∘

(e)

Figure 9 Ratio of plume radiation in the reduced scaling and full size rockets

Journal of Applied Mathematics 9

0

10000

20000

30000

40000

2 3 4 5 6

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

120582 (120583m)

I 120582W

(srmiddot120583

m)

(a)

0

10000

20000

30000

40000

I 120582W

(srmiddot120583

m)

2 3 4 5 6120582 (120583m)

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

(b)

Figure 10 Spectral radiation intensity of plume surface

0

2000

4000

6000

8000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

I 120582W

(srmiddot120583

m)

r = 05

(a)

0

10000

20000

30000

40000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

r = 1

I 120582W

(srmiddot120583

m)

(b)

Figure 11 Variation of radiation intensity of plume with zenith angles

Effect of radiance from the rear end also makes the directionwith maximum radiation intensity deflects toward the rearhemisphere which is remarkable for 119903 = 05 and themaximum radiation intensity arrives at 120579 = 50∘ in that caseThe colder section in the tail of plume is shorter and theradiance from the rear end surface would be bigger for asmaller scaling rocket

Figure 12 shows map of spectral radiance on 119909-119911 coordi-nate plane in plume flowof the 05 scaling solid rocket and thefull scaled ones for wavelength 120582 = 3 120583m and 5 120583m in 120579 = 0∘45∘ and 90∘ The radiance map of the two solid rockets hassimilar shape and close magnitude It is also found that the

high light area with a strong radiance is located in the centralstripe of the plume where temperature and concentrationof gases and Al

2O3particles are very high Compared the

width of the high light stripe in radiance maps of the twowavelengths it is narrower for 120582 = 3 120583m than 120582 = 5 120583mAs the radiation of Al

2O3particles plays the main role in

plume radiance of 3120583m the area with dense concentration ofparticles is smaller in radius than plume gases causing highradiance area centralizing in the plume in that wavelengthPlume gases are the major radiation composition for 120582 =5 120583m which has a bigger expansion angle and radius sizeand cause a wider stripe with high radiance Viewing the axial

10 Journal of Applied Mathematics

times102

z z z

z z z

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

r = 1 120582 = 3120583m 120579 = 0∘ r = 1 120582 = 3120583m 120579 = 45∘ r = 1 120582 = 3120583m 120579 = 90∘

r = 1 120582 = 5120583m 120579 = 0∘ r = 1 120582 = 5120583m 120579 = 45∘ r = 1 120582 = 5120583m 120579 = 90∘

1 2 3 4 5 6 7 8 9 10 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900times102

z

x

2 3 4 5 6 7 8 9 10 11 12

0

0 50 100 150 200 250 300

0 50 100 150 200 250 300 0 50 100 150 200 250 300

50 100 150 200 250 300 0

50

25

75

100

125

225

175

150

200

250

0 50 100 150 200 250

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

r = 05 120582 = 3120583m 120579 = 0∘ r = 05 120582 = 3120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

r = 05 120582 = 5120583m 120579 = 0∘ r = 05 120582 = 5120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

Figure 12 Map of spectral radiance on 119909-119911 coordinate plane

length of the high light stripe it is in the front part with halflength of the plume for the 05 scaling model but extendsfrom the front to end part of plume for the full scaled rocketmodel

8 Conclusion

Infrared radiation of plume from one full scaled solid rocketas well as the 01sim09 reducing scaled models in groundcondition has been investigated to study the similarity ofplumersquos infrared radiation for rocket models with differentscaling ratios Flowing field in the rocket and plume hasbeen computed with CFD code the omnidirectional andinfrared radiance on symmetric plane in the plume has beencalculated with the developed FVM codeThe research showsthat Al

2O3particle is the major radiation composition in the

rocket plumewhose scattering coefficient ismuch larger than

its absorption coefficient Ratio of radiation intensity fromreduced scaling plume to that of the full scaled one increaseswith the scaling ratio The power of the growth curve is 15for azimuth angle 120579 = 0∘ becomes 2 for 120579 = 40∘ and arrives at25 for 120579 = 120∘ and 160∘ There is good similarity betweenspectral variations of plumes from different scaling solidrockets and there are two peak spectra of radiation intensityin 27sim30 120583m and 43sim46 120583m Radiation intensity of theplume in rear hemisphere is bigger than that in the front andthe effect of radiance from the rear end surface makes thedirection with maximum radiation intensity deflects towardthe rear hemisphere

Conflict of Interests

There is not conflict of interests for all authors of this paper

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

2

0

z(m

)

5 10 15 20

x (m)

Figure 2 Grids for radiance calculation in the plume

where the subscript ldquo120582rdquo means spectral variable and thesuperscript ldquo1015840rdquo means directional variable The symbol Σ120581

119899

is the summed absorption coefficient of plume gases andAl2O3particles and 120590 is the summed scattering coefficient

of 8 groups of Al2O3particles which are simply summed

up mathematically The symbol S1015840 means direction of theincident radiation 1205961015840 means the solid angle in S1015840 directionand Φ(S S1015840) means the scattering phase function betweenS and S1015840 The above symbols 120581

119899 120590 and Φ are all spectral

variables while the notations ldquo120582rdquo are omitted in (1) forabbreviation

According to finite volume method (1) can be solvedto compute spectral radiance of plume by integrating theequation on a control volume Considering the plume geom-etry the cylindrical control volumes had been firstly usedfor solving (1) but it is found that numerical convergence ofthe developed coefficient matrix is very poor being as thearea difference between the two radial opposite faces of thecontrol volume For that reason the cuboid control volumehas finally been chosen in our study The new computationdomain for radiation computation has the same length withthe flowing domain in Figure 1 but it has a square formend with side length being equal to the outer radius of theplume The precision of plume radiance solution is mainlyaffected by flowing data temperature volume fraction ornumber concentration of plume gases and Al

2O3particles

Those flowing data do not change dramatically in the plumeas seen in Figure 1 so the mesh size for radiance solution canbe much larger than that used in CFD solution For plume ofthe full scaled solid rocket the meshing plot on lengthwisesection of the computing domain is shown in Figure 2 Thedivided grids number is 80 times 20 in the upper 119909-119911 planeFlowing data of control volume inside the plume takes thearithmetic average of data on CFD grids in the volume Thatof control volume outside the plume is set to zero In thatcase all control volumes outside the plumewould not actuallyinfluence the result of plume radiance but the integral RTEequation is uniform for all the control volumes and solutionconvergence can be improved

Spectral radiance of each control volume is consideredin a series of directions every direction is correspondingto a set of angle zenith angle and azimuth angle Thezenith angle 120579 is defined as the angle between the plumecentral line (119909-axis) and the direction vector and azimuthangle 120593 is defined as angle between the projected directionvectors on 119909-119911 coordinate plane and the 119911 axis For radianceon control volume 119875 in direction S1015840 integrating (1) withGaussian integral method radiance of 119875 can be related toradiance of its neighbored control volumes in that directionThe detailed derivation procedure can be found in [19] onlythe final integration equation is given here as follows

f

w

NF

R

S

EW P

rs

n

x

y

e

z

Figure 3 A control volume and the 6 neighbors

max [Δ119860

119908119863

119908 0] i1015840119875+ max [Δ119860

119890119863

119890 0] i1015840119875

+ max [Δ119860

119904119863

119904 0] i1015840119875+ max [Δ119860

119899119863

119899 0] i1015840119875

+ max [Δ119860

119903119863

119903 0] i1015840119875+ max [Δ119860

119891119863

119891 0] i1015840119875

+ (Σ120581

119899+ 120590)Δ119881

119875Δ120596

1015840i1015840119875

= max [minusΔ119860

119908119863

119908 0] i1015840119882

+ max [minusΔ119860

119890119863

119890 0] i1015840119864

+ max [minusΔ119860

119904119863

119904 0] i1015840119878+ max [minusΔ119860

119899119863

119899 0] i1015840119873

+ max [minusΔ119860

119903119863

119903 0] i1015840119877+ max [minusΔ119860

119891119863

119891 0] i1015840119865

+ [Σ (120581

119899) i1015840120582119887

+

120590

4120587

Σ (i1015840119875(S) Φ (S S1015840) Δ120596)]Δ119881

119875Δ120596

1015840

119863

119896= int

Δ1205961015840

(S sdot n119896) 119889120596

1015840 119896 = 119908 119890 119899 119904 119903 119891

Φ (S S1015840) =

int

Δ1205961015840int

Δ120596Φ(S S1015840) 119889120596

1015840119889120596

Δ120596

1015840Δ120596

Δ120596

1015840= int

120593119897+

120593119897minus

int

120579119897+

120579119897minus

sin 120579 119889120579 119889120593

(2)

where subscripts 119882 119864 119873 119878 119877 and 119865 mean the variables ofthe neighbored control volume and 119908 119890 119899 119904 119903 and 119891 meanthe variables on interface of neighboring volumes as shownin Figure 3 The symbol Δ119860

119896is the area of the interface n

119896is

external normal vector of interface and Δ119881

119875is the volume of

control volumeThe upper equations can be also written as the following

simplified form

119886

119875i1015840119875

= 119886

119882i1015840119882

+ 119886

119864i1015840119864+ 119886

119878i1015840119878+ 119886

119873i1015840119873

+ 119886

119877i1015840119877+ 119886

119865i1015840119865+ b119875

119886

119875= Σmax (Δ119860

119896119863

119896 0)

Journal of Applied Mathematics 5

+ (Σ120581

119899+ 120590 minus

120590

4120587

Φ (S1015840 S1015840) Δ120596

1015840)Δ119881

119875Δ120596

1015840

b119875

= [(Σ120581

119899) i1015840120582119887

+

120590

4120587

sum

S =S1015840i1015840119875(S) Φ (S S1015840) Δ120596]Δ119881

119875Δ120596

1015840

(3)

The compound integral RTE equations of all control volumeshave the form of vector equations

[A] [I] = [B] (4)

Equation (4) is a nonlinear system of equations as variableb119875in the right side is actually unknown and needs to be

determined by radiance of control volume 119875 in all the otherdirections S1015840 So the solution of (4) could not be completedwith any iteration algorithm in one loop A cyclic iterationalgorithm with modification of b

119875has been proposed in this

study In each cycle b119875is firstly calculated with the radiance

computed from the last cycle Then a Gauss iteration algo-rithm will be used to solve vector equations (4) to computea new group of radiance for each control volume The cycliciteration will go on till the following convergence limitationis met

max i119899119875minus i119899minus1119875

le 120576 (5)

Considering the symmetric characteristics of geometry andflowing data of the plume radiance of plume is also sym-metric So we do not need to solve the vector equations ofRTE on all control volumes in the plume but only thosecontrol volumes with centers on 119909-119911 coordinate plane havebeen chosen for calculation While radiance of neighboredcontrol volumes is also required in (4) those control volumesare not centering on the 119909-119911 coordinate plane and a radiancetransformation algorithm on symmetric control volumes hasbeen proposed in our study to calculate the radiance of othercontrol volumes in the plume The radiance transformationalgorithm is derived based on rotation rules of vectors seenin Figure 4

If the radiance of control volume 119875 in direction S1is

i1015840119875(S1) the radiance of control volume 119874 in direction S

2is

equal to i1015840119875(S1)when the following vector equation is satisfied

p2= S2+ r2 (6)

4 Calculating Radiation Intensity fromthe Plume Surface

In the cases of studying base heating or heat protection of thesolid rocket spectral radiation intensity on the whole surfaceof the plume is concerned Spectral radiation intensity is alsoa directional variable which is the integration of spectralradiance i1015840

120582(S) on surface of bounding control volumes

in direction S If radiation intensity in some waveband

z

PQ

y

1198511

1198261

1198262

1198262

1198491

11984921198512

120575120575

120579

120579

Figure 4 Radiance transformation on symmetric control volumes

is concerned as for the 2sim6120583m band integration on thewavelength is also needed

119868

120582= sum

119895119860

i1015840120582(S) eS sdot n

Δ119860Δ119860 eS sdot n

Δ119860gt 0

119868 = sum

119895120582

sum

119895119860

i1015840 (S) eS sdot nΔ119860

Δ119860Δ120582

(7)

where 119895

119860is number of control volume surfaces on boundary

of the plume eS is unit vector of the radiance direction andnΔ119860

is unit vector of external normal on the surface

5 Calculation of Spectral Property ofPlume Gases and Al2O3 Particles

The3major radiation gases of the solid rocket plume areH2O

CO2 and CO To calculate the spectral absorption coefficient

of each plume gas the standard absorption coefficient 120581120582STP

at several temperatures and 1 atm pressure in [20 21] has beenused For plume gas with pressure being 119875 and temperaturebeing 119879 absorption coefficient can be modified from 120581

120582STPas follows

120581

120582=

119875

10

5

273

119879

120581

120582STP (8)

Absorption coefficient of the plume gases mixture is calcu-lated with the WSGG model and the weight is the volumefraction of each gas If the volume fraction of one gas is 119891then the absorption coefficient of the gases mixture can bewritten with the WSGG model as

120581

120582119892= 120581

120582H2O119891H

2O + 120581

120582CO2

119891CO2

+ 120581

120582CO119891CO (9)

The calculated absorption coefficient for plume gases H2O

CO2 and CO and the gases mixture in center of front end of

the plume have been shown in Figure 5 One can find thatabsorption coefficient of CO

2and CO is comparatively much

larger than that of CO is 42sim55 120583m the peak spectrum ofCO2is 42sim5 120583m and absorption coefficient of H

2O is much

smaller Absorption coefficient of the gases mixture appearsas an accumulated effect of the three gases

6 Journal of Applied Mathematics

0

1

2

3

4

5

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

kH2OkCO2kCO

kHCLkgases

Figure 5 Calculated absorption coefficient of plume gases withWSGG model

6 Calculation of the Spectral Property ofAl2O3 Particles

In the calculation of spectral properties of Al2O3particles

with Mie scattering model particle size number concen-tration and complex refractive index must be determinedfirstly Asmentioned above size andnumber concentration ofAl2O3particles have been calculated in plume flowing com-

putation The complex refractive index of Al2O3particles

m = 119899 + i119896 takes formulas recommended by Reed et al [12]

119896 = 466 times 10

minus4120582

133119879

15 exp [minus

29420

119879

]

119899 = 175 cos (6120582) (10)

For a single Al2O3particle spectral properties including

the scattering cross section 119862

119904120582 attenuation cross section

119862

119890120582 scattering factor 119876

119904120582 attenuation factor 119876

119890120582 and the

scattering phase function Φ

120582can be computed with Mie

scattering model [13] as

119862

119904120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11a)

119862

119890120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11b)

119876

119904120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11c)

119876

119890120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11d)

Φ

120582(120579) =

2

119876

119904120582120594

2[

1003816

1003816

1003816

1003816

119878

1

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119878

2

1003816

1003816

1003816

1003816

2

] (11e)

where 119886

119899and 119887

119899are Mie scattering coefficients and 119878

1

and 119878

2are amplitude functions Computation of the above

0

1

2

3

4

5

6

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

All

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

Figure 6 Calculated absorption coefficient of Al2O3particles with

Mie model

coefficients and functions has been illustrated in detail in [13]which is not repeated in this work

As for the spectral properties of the particle clouds thesingle and independent effect of each particle is assumedand all the spectral properties of the particle clouds can becomputed as the mathematic sum of all particles

The calculated absorption coefficients of 6 groups ofAl2O3particles with different diameters and the total value

of particles cloud in the center of plumersquos front end havebeen shown in Figure 6 That of the other two groupswith 119889 = 4 120583m 12 120583m is not included in the figure as theirconcentration is zero at front end of the plume One cansee that absorption coefficient of Al

2O3particles does not

change significantly with wavelength like that of plume gasesThe three groups of Al

2O3particles with smaller diameters

(119889 = 4 120583m 6 120583m 8 120583m) have larger absorption coefficientCompared with Figure 5 it is found that the absorptioncoefficient of particles cloud ismuch larger than that of plumegases so Al

2O3particles are the major radiation composition

in the rocket plumeThe calculated scattering coefficients of the 6 groups of

Al2O3particles with different diameters and total value of

particles cloud in the center of plumersquos front face have beenshown in Figure 7 One can find that scattering coefficientfor Al

2O3particles with 119889 = 4 120583m is most strong which

is medium for particles with 119889 = 6 120583m and is very smallfor extra particles Compared with Figure 6 it is found thatscattering coefficient of particles cloud is much bigger thanthe absorption coefficient and the scattering coefficient willgrow up with the increasing wavelength

The calculated scattering phase functions of Al2O3parti-

cles cloud for different wavelengths in the center of plumersquosfront end have been shown in Figure 8The scattering angle 120579

in the figure means the angle between the incident radianceand scattering direction It can be seen that the scatteringphase function in forward directions is much larger than

Journal of Applied Mathematics 7

0

2

4

6

8

10

12

2 3 4 5 6

All

120582 (120583m)

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

120590(c

mminus

1 )

Figure 7 Calculated scattering coefficient of Al2O3particles with

Mie model

that in rear directions and scattering phase function instraight forward direction is the largest among all the fivewavelengths Comparing the results for the five wavelengthsin straight forward direction the smaller wavelength has alarger forward scattering phase function

7 Results and Discussion

The calculation method of spectral property of plume gasesand Al

2O3particles and infrared radiation intensity has been

verified with a reduced scaling solid rocket in a vacuumchamber Infrared radiation intensity in wavebands 27sim295 120583m and 42sim445 120583m at three zenith angles 120579 = 60∘90∘ and 120∘ have been calculated and compared with testresults The maximum difference between calculated and testresults is 209 which arrives at 120579 = 90∘ and the minimumdifference is 104 which arrives at 120579 = 60∘ [22]

Studying the similarities of plumersquos radiation intensityfrom solid rocket with different scale and ratio of plumeradiation between the reduced scaling and full size rockets inwavelengths 120582 = 2 120583m 3 120583m 4 120583m and 5 120583m and waveband2sim6120583m have been calculated and illustrated in Figure 9 Thehorizontal ordinate 119903 in the figure means the scaling ratioof rocket model and the vertical ordinate 119868119868

0means the

ratio of plumersquos radiation intensityThe three dashed lines arecorresponding to the 15 2 and 25 power function of scalingratio Results of 120579 = 0∘ 40∘ 80∘ 120∘ and 160∘ with 120593 = 30∘have been investigated It can be seen in the figure that ratioof radiation intensity increases with the increasing scalingratio for all the four wavelengths and waveband 2sim6 120583mTheincreasing rule of radiation intensity ratio for 120579 = 0∘ gets closeto the curve of 15 power mostly which gets closer to the 2-power function of scaling ratio for 120579 = 40∘ and comes up toreach 25 power of scaling ratio for 120579 = 120∘ and 160∘ This iscaused by changing the length of high radiance section and

01

1

10

100

0 30 60 90 120 150 180

Phas

e fun

ctio

nΦ

Scattering angle 120579

120582 = 2120583m120582 = 3120583m120582 = 4120583m

120582 = 5120583m120582 = 6120583m

Figure 8 Calculated scattering phase function of Al2O3particles

with Mie model

radiance attenuation of the cold gases in tail section of theplume As plumersquos radiance mainly comes from the centralstripe which has a higher temperature and concentration forboth gases and Al

2O3particles Radiance attenuation of the

tail cold gases increases more rapidly than the projected areaon the surface of the plume with the growth of the scalingratio making growing rate of 119868119868

0being smaller than that of

surface area the latter has a growing rate of 1199032 in small zenithangle With the increase of zenith angle the effect of radianceattenuation decreases greatly and causes the ratio of radiationintensity increasing to reach and surpass 119903

2In the viewing of spectral characteristics for solid rocket

plume with different scaling ratios the variation of radiationintensity with wavelength for the 9 reduced scaling rocketmodels and the full scaled one in two directions 120579 =30∘ and 90∘ with 120593 = 30∘ has been plotted in Figure 10One can see good similarity between spectral variations ofdifferent scaling rocket models There are two peak spectraof radiation intensity 27sim30 120583mand 43sim46 120583m Comparedwith Figure 5 it is found that the former spectrum is thefeature radiative spectrum of H

2O the latter is of CO

2and

CO Besides in the two peak spectra spectral radiationintensity for plume of the 10 rockets decreases with theincreasing wavelength in 2sim6 120583mwaveband Since the Al

2O3

particle is the major radiation composition in the plumewhich has the maximum radiation for 120582 lt 20 120583m as particlethe temperature is higher than 2500K

The spectral radiation intensity of plume in all the zenithangles with 120593 = 30∘ for the 05 scaling rocket model and thefull scaled one has been shown in Figure 11 results of wave-length 120582 = 2 120583m 3 120583m and 4 120583m have been plotted With theincrease of the zenith angle radiation intensity of the plumefirstly increases arrives at a maximum value in a middleangle and decreases after that angle since the projected areaon plume surface changes with the zenith angle Comparingradiation intensity of plume in the front and rear hemisphereitrsquos found that is stronger in the rear hemisphere As read endsurface of plume has a strong radiance in the rear hemisphere

8 Journal of Applied Mathematics

0

03

06

09

12

15

0 02 04 06 08 1

II 0

120579 = 0∘ 120593 = 30∘

(a)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 40∘ 120593 = 30∘

(b)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 80∘ 120593 = 30∘

(c)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 120∘ 120593 = 30∘

(d)

0

02

04

06

08

1

0 02 04 06 08 1

120582 = 2120583m120582 = 3120583m120582 = 4120583m120582 = 5120583m

120582 = 2sim6120583mr15

r2

r25

II 0

120579 = 160∘ 120593 = 30∘

(e)

Figure 9 Ratio of plume radiation in the reduced scaling and full size rockets

Journal of Applied Mathematics 9

0

10000

20000

30000

40000

2 3 4 5 6

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

120582 (120583m)

I 120582W

(srmiddot120583

m)

(a)

0

10000

20000

30000

40000

I 120582W

(srmiddot120583

m)

2 3 4 5 6120582 (120583m)

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

(b)

Figure 10 Spectral radiation intensity of plume surface

0

2000

4000

6000

8000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

I 120582W

(srmiddot120583

m)

r = 05

(a)

0

10000

20000

30000

40000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

r = 1

I 120582W

(srmiddot120583

m)

(b)

Figure 11 Variation of radiation intensity of plume with zenith angles

Effect of radiance from the rear end also makes the directionwith maximum radiation intensity deflects toward the rearhemisphere which is remarkable for 119903 = 05 and themaximum radiation intensity arrives at 120579 = 50∘ in that caseThe colder section in the tail of plume is shorter and theradiance from the rear end surface would be bigger for asmaller scaling rocket

Figure 12 shows map of spectral radiance on 119909-119911 coordi-nate plane in plume flowof the 05 scaling solid rocket and thefull scaled ones for wavelength 120582 = 3 120583m and 5 120583m in 120579 = 0∘45∘ and 90∘ The radiance map of the two solid rockets hassimilar shape and close magnitude It is also found that the

high light area with a strong radiance is located in the centralstripe of the plume where temperature and concentrationof gases and Al

2O3particles are very high Compared the

width of the high light stripe in radiance maps of the twowavelengths it is narrower for 120582 = 3 120583m than 120582 = 5 120583mAs the radiation of Al

2O3particles plays the main role in

plume radiance of 3120583m the area with dense concentration ofparticles is smaller in radius than plume gases causing highradiance area centralizing in the plume in that wavelengthPlume gases are the major radiation composition for 120582 =5 120583m which has a bigger expansion angle and radius sizeand cause a wider stripe with high radiance Viewing the axial

10 Journal of Applied Mathematics

times102

z z z

z z z

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

r = 1 120582 = 3120583m 120579 = 0∘ r = 1 120582 = 3120583m 120579 = 45∘ r = 1 120582 = 3120583m 120579 = 90∘

r = 1 120582 = 5120583m 120579 = 0∘ r = 1 120582 = 5120583m 120579 = 45∘ r = 1 120582 = 5120583m 120579 = 90∘

1 2 3 4 5 6 7 8 9 10 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900times102

z

x

2 3 4 5 6 7 8 9 10 11 12

0

0 50 100 150 200 250 300

0 50 100 150 200 250 300 0 50 100 150 200 250 300

50 100 150 200 250 300 0

50

25

75

100

125

225

175

150

200

250

0 50 100 150 200 250

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

r = 05 120582 = 3120583m 120579 = 0∘ r = 05 120582 = 3120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

r = 05 120582 = 5120583m 120579 = 0∘ r = 05 120582 = 5120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

Figure 12 Map of spectral radiance on 119909-119911 coordinate plane

length of the high light stripe it is in the front part with halflength of the plume for the 05 scaling model but extendsfrom the front to end part of plume for the full scaled rocketmodel

8 Conclusion

Infrared radiation of plume from one full scaled solid rocketas well as the 01sim09 reducing scaled models in groundcondition has been investigated to study the similarity ofplumersquos infrared radiation for rocket models with differentscaling ratios Flowing field in the rocket and plume hasbeen computed with CFD code the omnidirectional andinfrared radiance on symmetric plane in the plume has beencalculated with the developed FVM codeThe research showsthat Al

2O3particle is the major radiation composition in the

rocket plumewhose scattering coefficient ismuch larger than

its absorption coefficient Ratio of radiation intensity fromreduced scaling plume to that of the full scaled one increaseswith the scaling ratio The power of the growth curve is 15for azimuth angle 120579 = 0∘ becomes 2 for 120579 = 40∘ and arrives at25 for 120579 = 120∘ and 160∘ There is good similarity betweenspectral variations of plumes from different scaling solidrockets and there are two peak spectra of radiation intensityin 27sim30 120583m and 43sim46 120583m Radiation intensity of theplume in rear hemisphere is bigger than that in the front andthe effect of radiance from the rear end surface makes thedirection with maximum radiation intensity deflects towardthe rear hemisphere

Conflict of Interests

There is not conflict of interests for all authors of this paper

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

+ (Σ120581

119899+ 120590 minus

120590

4120587

Φ (S1015840 S1015840) Δ120596

1015840)Δ119881

119875Δ120596

1015840

b119875

= [(Σ120581

119899) i1015840120582119887

+

120590

4120587

sum

S =S1015840i1015840119875(S) Φ (S S1015840) Δ120596]Δ119881

119875Δ120596

1015840

(3)

The compound integral RTE equations of all control volumeshave the form of vector equations

[A] [I] = [B] (4)

Equation (4) is a nonlinear system of equations as variableb119875in the right side is actually unknown and needs to be

determined by radiance of control volume 119875 in all the otherdirections S1015840 So the solution of (4) could not be completedwith any iteration algorithm in one loop A cyclic iterationalgorithm with modification of b

119875has been proposed in this

study In each cycle b119875is firstly calculated with the radiance

computed from the last cycle Then a Gauss iteration algo-rithm will be used to solve vector equations (4) to computea new group of radiance for each control volume The cycliciteration will go on till the following convergence limitationis met

max i119899119875minus i119899minus1119875

le 120576 (5)

Considering the symmetric characteristics of geometry andflowing data of the plume radiance of plume is also sym-metric So we do not need to solve the vector equations ofRTE on all control volumes in the plume but only thosecontrol volumes with centers on 119909-119911 coordinate plane havebeen chosen for calculation While radiance of neighboredcontrol volumes is also required in (4) those control volumesare not centering on the 119909-119911 coordinate plane and a radiancetransformation algorithm on symmetric control volumes hasbeen proposed in our study to calculate the radiance of othercontrol volumes in the plume The radiance transformationalgorithm is derived based on rotation rules of vectors seenin Figure 4

If the radiance of control volume 119875 in direction S1is

i1015840119875(S1) the radiance of control volume 119874 in direction S

2is

equal to i1015840119875(S1)when the following vector equation is satisfied

p2= S2+ r2 (6)

4 Calculating Radiation Intensity fromthe Plume Surface

In the cases of studying base heating or heat protection of thesolid rocket spectral radiation intensity on the whole surfaceof the plume is concerned Spectral radiation intensity is alsoa directional variable which is the integration of spectralradiance i1015840

120582(S) on surface of bounding control volumes

in direction S If radiation intensity in some waveband

z

PQ

y

1198511

1198261

1198262

1198262

1198491

11984921198512

120575120575

120579

120579

Figure 4 Radiance transformation on symmetric control volumes

is concerned as for the 2sim6120583m band integration on thewavelength is also needed

119868

120582= sum

119895119860

i1015840120582(S) eS sdot n

Δ119860Δ119860 eS sdot n

Δ119860gt 0

119868 = sum

119895120582

sum

119895119860

i1015840 (S) eS sdot nΔ119860

Δ119860Δ120582

(7)

where 119895

119860is number of control volume surfaces on boundary

of the plume eS is unit vector of the radiance direction andnΔ119860

is unit vector of external normal on the surface

5 Calculation of Spectral Property ofPlume Gases and Al2O3 Particles

The3major radiation gases of the solid rocket plume areH2O

CO2 and CO To calculate the spectral absorption coefficient

of each plume gas the standard absorption coefficient 120581120582STP

at several temperatures and 1 atm pressure in [20 21] has beenused For plume gas with pressure being 119875 and temperaturebeing 119879 absorption coefficient can be modified from 120581

120582STPas follows

120581

120582=

119875

10

5

273

119879

120581

120582STP (8)

Absorption coefficient of the plume gases mixture is calcu-lated with the WSGG model and the weight is the volumefraction of each gas If the volume fraction of one gas is 119891then the absorption coefficient of the gases mixture can bewritten with the WSGG model as

120581

120582119892= 120581

120582H2O119891H

2O + 120581

120582CO2

119891CO2

+ 120581

120582CO119891CO (9)

The calculated absorption coefficient for plume gases H2O

CO2 and CO and the gases mixture in center of front end of

the plume have been shown in Figure 5 One can find thatabsorption coefficient of CO

2and CO is comparatively much

larger than that of CO is 42sim55 120583m the peak spectrum ofCO2is 42sim5 120583m and absorption coefficient of H

2O is much

smaller Absorption coefficient of the gases mixture appearsas an accumulated effect of the three gases

6 Journal of Applied Mathematics

0

1

2

3

4

5

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

kH2OkCO2kCO

kHCLkgases

Figure 5 Calculated absorption coefficient of plume gases withWSGG model

6 Calculation of the Spectral Property ofAl2O3 Particles

In the calculation of spectral properties of Al2O3particles

with Mie scattering model particle size number concen-tration and complex refractive index must be determinedfirstly Asmentioned above size andnumber concentration ofAl2O3particles have been calculated in plume flowing com-

putation The complex refractive index of Al2O3particles

m = 119899 + i119896 takes formulas recommended by Reed et al [12]

119896 = 466 times 10

minus4120582

133119879

15 exp [minus

29420

119879

]

119899 = 175 cos (6120582) (10)

For a single Al2O3particle spectral properties including

the scattering cross section 119862

119904120582 attenuation cross section

119862

119890120582 scattering factor 119876

119904120582 attenuation factor 119876

119890120582 and the

scattering phase function Φ

120582can be computed with Mie

scattering model [13] as

119862

119904120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11a)

119862

119890120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11b)

119876

119904120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11c)

119876

119890120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11d)

Φ

120582(120579) =

2

119876

119904120582120594

2[

1003816

1003816

1003816

1003816

119878

1

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119878

2

1003816

1003816

1003816

1003816

2

] (11e)

where 119886

119899and 119887

119899are Mie scattering coefficients and 119878

1

and 119878

2are amplitude functions Computation of the above

0

1

2

3

4

5

6

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

All

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

Figure 6 Calculated absorption coefficient of Al2O3particles with

Mie model

coefficients and functions has been illustrated in detail in [13]which is not repeated in this work

As for the spectral properties of the particle clouds thesingle and independent effect of each particle is assumedand all the spectral properties of the particle clouds can becomputed as the mathematic sum of all particles

The calculated absorption coefficients of 6 groups ofAl2O3particles with different diameters and the total value

of particles cloud in the center of plumersquos front end havebeen shown in Figure 6 That of the other two groupswith 119889 = 4 120583m 12 120583m is not included in the figure as theirconcentration is zero at front end of the plume One cansee that absorption coefficient of Al

2O3particles does not

change significantly with wavelength like that of plume gasesThe three groups of Al

2O3particles with smaller diameters

(119889 = 4 120583m 6 120583m 8 120583m) have larger absorption coefficientCompared with Figure 5 it is found that the absorptioncoefficient of particles cloud ismuch larger than that of plumegases so Al

2O3particles are the major radiation composition

in the rocket plumeThe calculated scattering coefficients of the 6 groups of

Al2O3particles with different diameters and total value of

particles cloud in the center of plumersquos front face have beenshown in Figure 7 One can find that scattering coefficientfor Al

2O3particles with 119889 = 4 120583m is most strong which

is medium for particles with 119889 = 6 120583m and is very smallfor extra particles Compared with Figure 6 it is found thatscattering coefficient of particles cloud is much bigger thanthe absorption coefficient and the scattering coefficient willgrow up with the increasing wavelength

The calculated scattering phase functions of Al2O3parti-

cles cloud for different wavelengths in the center of plumersquosfront end have been shown in Figure 8The scattering angle 120579

in the figure means the angle between the incident radianceand scattering direction It can be seen that the scatteringphase function in forward directions is much larger than

Journal of Applied Mathematics 7

0

2

4

6

8

10

12

2 3 4 5 6

All

120582 (120583m)

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

120590(c

mminus

1 )

Figure 7 Calculated scattering coefficient of Al2O3particles with

Mie model

that in rear directions and scattering phase function instraight forward direction is the largest among all the fivewavelengths Comparing the results for the five wavelengthsin straight forward direction the smaller wavelength has alarger forward scattering phase function

7 Results and Discussion

The calculation method of spectral property of plume gasesand Al

2O3particles and infrared radiation intensity has been

verified with a reduced scaling solid rocket in a vacuumchamber Infrared radiation intensity in wavebands 27sim295 120583m and 42sim445 120583m at three zenith angles 120579 = 60∘90∘ and 120∘ have been calculated and compared with testresults The maximum difference between calculated and testresults is 209 which arrives at 120579 = 90∘ and the minimumdifference is 104 which arrives at 120579 = 60∘ [22]

Studying the similarities of plumersquos radiation intensityfrom solid rocket with different scale and ratio of plumeradiation between the reduced scaling and full size rockets inwavelengths 120582 = 2 120583m 3 120583m 4 120583m and 5 120583m and waveband2sim6120583m have been calculated and illustrated in Figure 9 Thehorizontal ordinate 119903 in the figure means the scaling ratioof rocket model and the vertical ordinate 119868119868

0means the

ratio of plumersquos radiation intensityThe three dashed lines arecorresponding to the 15 2 and 25 power function of scalingratio Results of 120579 = 0∘ 40∘ 80∘ 120∘ and 160∘ with 120593 = 30∘have been investigated It can be seen in the figure that ratioof radiation intensity increases with the increasing scalingratio for all the four wavelengths and waveband 2sim6 120583mTheincreasing rule of radiation intensity ratio for 120579 = 0∘ gets closeto the curve of 15 power mostly which gets closer to the 2-power function of scaling ratio for 120579 = 40∘ and comes up toreach 25 power of scaling ratio for 120579 = 120∘ and 160∘ This iscaused by changing the length of high radiance section and

01

1

10

100

0 30 60 90 120 150 180

Phas

e fun

ctio

nΦ

Scattering angle 120579

120582 = 2120583m120582 = 3120583m120582 = 4120583m

120582 = 5120583m120582 = 6120583m

Figure 8 Calculated scattering phase function of Al2O3particles

with Mie model

radiance attenuation of the cold gases in tail section of theplume As plumersquos radiance mainly comes from the centralstripe which has a higher temperature and concentration forboth gases and Al

2O3particles Radiance attenuation of the

tail cold gases increases more rapidly than the projected areaon the surface of the plume with the growth of the scalingratio making growing rate of 119868119868

0being smaller than that of

surface area the latter has a growing rate of 1199032 in small zenithangle With the increase of zenith angle the effect of radianceattenuation decreases greatly and causes the ratio of radiationintensity increasing to reach and surpass 119903

2In the viewing of spectral characteristics for solid rocket

plume with different scaling ratios the variation of radiationintensity with wavelength for the 9 reduced scaling rocketmodels and the full scaled one in two directions 120579 =30∘ and 90∘ with 120593 = 30∘ has been plotted in Figure 10One can see good similarity between spectral variations ofdifferent scaling rocket models There are two peak spectraof radiation intensity 27sim30 120583mand 43sim46 120583m Comparedwith Figure 5 it is found that the former spectrum is thefeature radiative spectrum of H

2O the latter is of CO

2and

CO Besides in the two peak spectra spectral radiationintensity for plume of the 10 rockets decreases with theincreasing wavelength in 2sim6 120583mwaveband Since the Al

2O3

particle is the major radiation composition in the plumewhich has the maximum radiation for 120582 lt 20 120583m as particlethe temperature is higher than 2500K

The spectral radiation intensity of plume in all the zenithangles with 120593 = 30∘ for the 05 scaling rocket model and thefull scaled one has been shown in Figure 11 results of wave-length 120582 = 2 120583m 3 120583m and 4 120583m have been plotted With theincrease of the zenith angle radiation intensity of the plumefirstly increases arrives at a maximum value in a middleangle and decreases after that angle since the projected areaon plume surface changes with the zenith angle Comparingradiation intensity of plume in the front and rear hemisphereitrsquos found that is stronger in the rear hemisphere As read endsurface of plume has a strong radiance in the rear hemisphere

8 Journal of Applied Mathematics

0

03

06

09

12

15

0 02 04 06 08 1

II 0

120579 = 0∘ 120593 = 30∘

(a)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 40∘ 120593 = 30∘

(b)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 80∘ 120593 = 30∘

(c)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 120∘ 120593 = 30∘

(d)

0

02

04

06

08

1

0 02 04 06 08 1

120582 = 2120583m120582 = 3120583m120582 = 4120583m120582 = 5120583m

120582 = 2sim6120583mr15

r2

r25

II 0

120579 = 160∘ 120593 = 30∘

(e)

Figure 9 Ratio of plume radiation in the reduced scaling and full size rockets

Journal of Applied Mathematics 9

0

10000

20000

30000

40000

2 3 4 5 6

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

120582 (120583m)

I 120582W

(srmiddot120583

m)

(a)

0

10000

20000

30000

40000

I 120582W

(srmiddot120583

m)

2 3 4 5 6120582 (120583m)

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

(b)

Figure 10 Spectral radiation intensity of plume surface

0

2000

4000

6000

8000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

I 120582W

(srmiddot120583

m)

r = 05

(a)

0

10000

20000

30000

40000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

r = 1

I 120582W

(srmiddot120583

m)

(b)

Figure 11 Variation of radiation intensity of plume with zenith angles

Effect of radiance from the rear end also makes the directionwith maximum radiation intensity deflects toward the rearhemisphere which is remarkable for 119903 = 05 and themaximum radiation intensity arrives at 120579 = 50∘ in that caseThe colder section in the tail of plume is shorter and theradiance from the rear end surface would be bigger for asmaller scaling rocket

Figure 12 shows map of spectral radiance on 119909-119911 coordi-nate plane in plume flowof the 05 scaling solid rocket and thefull scaled ones for wavelength 120582 = 3 120583m and 5 120583m in 120579 = 0∘45∘ and 90∘ The radiance map of the two solid rockets hassimilar shape and close magnitude It is also found that the

high light area with a strong radiance is located in the centralstripe of the plume where temperature and concentrationof gases and Al

2O3particles are very high Compared the

width of the high light stripe in radiance maps of the twowavelengths it is narrower for 120582 = 3 120583m than 120582 = 5 120583mAs the radiation of Al

2O3particles plays the main role in

plume radiance of 3120583m the area with dense concentration ofparticles is smaller in radius than plume gases causing highradiance area centralizing in the plume in that wavelengthPlume gases are the major radiation composition for 120582 =5 120583m which has a bigger expansion angle and radius sizeand cause a wider stripe with high radiance Viewing the axial

10 Journal of Applied Mathematics

times102

z z z

z z z

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

r = 1 120582 = 3120583m 120579 = 0∘ r = 1 120582 = 3120583m 120579 = 45∘ r = 1 120582 = 3120583m 120579 = 90∘

r = 1 120582 = 5120583m 120579 = 0∘ r = 1 120582 = 5120583m 120579 = 45∘ r = 1 120582 = 5120583m 120579 = 90∘

1 2 3 4 5 6 7 8 9 10 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900times102

z

x

2 3 4 5 6 7 8 9 10 11 12

0

0 50 100 150 200 250 300

0 50 100 150 200 250 300 0 50 100 150 200 250 300

50 100 150 200 250 300 0

50

25

75

100

125

225

175

150

200

250

0 50 100 150 200 250

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

r = 05 120582 = 3120583m 120579 = 0∘ r = 05 120582 = 3120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

r = 05 120582 = 5120583m 120579 = 0∘ r = 05 120582 = 5120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

Figure 12 Map of spectral radiance on 119909-119911 coordinate plane

length of the high light stripe it is in the front part with halflength of the plume for the 05 scaling model but extendsfrom the front to end part of plume for the full scaled rocketmodel

8 Conclusion

Infrared radiation of plume from one full scaled solid rocketas well as the 01sim09 reducing scaled models in groundcondition has been investigated to study the similarity ofplumersquos infrared radiation for rocket models with differentscaling ratios Flowing field in the rocket and plume hasbeen computed with CFD code the omnidirectional andinfrared radiance on symmetric plane in the plume has beencalculated with the developed FVM codeThe research showsthat Al

2O3particle is the major radiation composition in the

rocket plumewhose scattering coefficient ismuch larger than

its absorption coefficient Ratio of radiation intensity fromreduced scaling plume to that of the full scaled one increaseswith the scaling ratio The power of the growth curve is 15for azimuth angle 120579 = 0∘ becomes 2 for 120579 = 40∘ and arrives at25 for 120579 = 120∘ and 160∘ There is good similarity betweenspectral variations of plumes from different scaling solidrockets and there are two peak spectra of radiation intensityin 27sim30 120583m and 43sim46 120583m Radiation intensity of theplume in rear hemisphere is bigger than that in the front andthe effect of radiance from the rear end surface makes thedirection with maximum radiation intensity deflects towardthe rear hemisphere

Conflict of Interests

There is not conflict of interests for all authors of this paper

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

0

1

2

3

4

5

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

kH2OkCO2kCO

kHCLkgases

Figure 5 Calculated absorption coefficient of plume gases withWSGG model

6 Calculation of the Spectral Property ofAl2O3 Particles

In the calculation of spectral properties of Al2O3particles

with Mie scattering model particle size number concen-tration and complex refractive index must be determinedfirstly Asmentioned above size andnumber concentration ofAl2O3particles have been calculated in plume flowing com-

putation The complex refractive index of Al2O3particles

m = 119899 + i119896 takes formulas recommended by Reed et al [12]

119896 = 466 times 10

minus4120582

133119879

15 exp [minus

29420

119879

]

119899 = 175 cos (6120582) (10)

For a single Al2O3particle spectral properties including

the scattering cross section 119862

119904120582 attenuation cross section

119862

119890120582 scattering factor 119876

119904120582 attenuation factor 119876

119890120582 and the

scattering phase function Φ

120582can be computed with Mie

scattering model [13] as

119862

119904120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11a)

119862

119890120582=

120582

2

2120587

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11b)

119876

119904120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1) [

1003816

1003816

1003816

1003816

119886

119899

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119887

119899

1003816

1003816

1003816

1003816

2

] (11c)

119876

119890120582=

2

120594

2

infin

sum

119899=1

(2119899 + 1)Re (119886

119899+ 119887

119899) (11d)

Φ

120582(120579) =

2

119876

119904120582120594

2[

1003816

1003816

1003816

1003816

119878

1

1003816

1003816

1003816

1003816

2

+

1003816

1003816

1003816

1003816

119878

2

1003816

1003816

1003816

1003816

2

] (11e)

where 119886

119899and 119887

119899are Mie scattering coefficients and 119878

1

and 119878

2are amplitude functions Computation of the above

0

1

2

3

4

5

6

2 3 4 5 6120582 (120583m)

120581(c

mminus1)

All

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

Figure 6 Calculated absorption coefficient of Al2O3particles with

Mie model

coefficients and functions has been illustrated in detail in [13]which is not repeated in this work

As for the spectral properties of the particle clouds thesingle and independent effect of each particle is assumedand all the spectral properties of the particle clouds can becomputed as the mathematic sum of all particles

The calculated absorption coefficients of 6 groups ofAl2O3particles with different diameters and the total value

of particles cloud in the center of plumersquos front end havebeen shown in Figure 6 That of the other two groupswith 119889 = 4 120583m 12 120583m is not included in the figure as theirconcentration is zero at front end of the plume One cansee that absorption coefficient of Al

2O3particles does not

change significantly with wavelength like that of plume gasesThe three groups of Al

2O3particles with smaller diameters

(119889 = 4 120583m 6 120583m 8 120583m) have larger absorption coefficientCompared with Figure 5 it is found that the absorptioncoefficient of particles cloud ismuch larger than that of plumegases so Al

2O3particles are the major radiation composition

in the rocket plumeThe calculated scattering coefficients of the 6 groups of

Al2O3particles with different diameters and total value of

particles cloud in the center of plumersquos front face have beenshown in Figure 7 One can find that scattering coefficientfor Al

2O3particles with 119889 = 4 120583m is most strong which

is medium for particles with 119889 = 6 120583m and is very smallfor extra particles Compared with Figure 6 it is found thatscattering coefficient of particles cloud is much bigger thanthe absorption coefficient and the scattering coefficient willgrow up with the increasing wavelength

The calculated scattering phase functions of Al2O3parti-

cles cloud for different wavelengths in the center of plumersquosfront end have been shown in Figure 8The scattering angle 120579

in the figure means the angle between the incident radianceand scattering direction It can be seen that the scatteringphase function in forward directions is much larger than

Journal of Applied Mathematics 7

0

2

4

6

8

10

12

2 3 4 5 6

All

120582 (120583m)

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

120590(c

mminus

1 )

Figure 7 Calculated scattering coefficient of Al2O3particles with

Mie model

that in rear directions and scattering phase function instraight forward direction is the largest among all the fivewavelengths Comparing the results for the five wavelengthsin straight forward direction the smaller wavelength has alarger forward scattering phase function

7 Results and Discussion

The calculation method of spectral property of plume gasesand Al

2O3particles and infrared radiation intensity has been

verified with a reduced scaling solid rocket in a vacuumchamber Infrared radiation intensity in wavebands 27sim295 120583m and 42sim445 120583m at three zenith angles 120579 = 60∘90∘ and 120∘ have been calculated and compared with testresults The maximum difference between calculated and testresults is 209 which arrives at 120579 = 90∘ and the minimumdifference is 104 which arrives at 120579 = 60∘ [22]

Studying the similarities of plumersquos radiation intensityfrom solid rocket with different scale and ratio of plumeradiation between the reduced scaling and full size rockets inwavelengths 120582 = 2 120583m 3 120583m 4 120583m and 5 120583m and waveband2sim6120583m have been calculated and illustrated in Figure 9 Thehorizontal ordinate 119903 in the figure means the scaling ratioof rocket model and the vertical ordinate 119868119868

0means the

ratio of plumersquos radiation intensityThe three dashed lines arecorresponding to the 15 2 and 25 power function of scalingratio Results of 120579 = 0∘ 40∘ 80∘ 120∘ and 160∘ with 120593 = 30∘have been investigated It can be seen in the figure that ratioof radiation intensity increases with the increasing scalingratio for all the four wavelengths and waveband 2sim6 120583mTheincreasing rule of radiation intensity ratio for 120579 = 0∘ gets closeto the curve of 15 power mostly which gets closer to the 2-power function of scaling ratio for 120579 = 40∘ and comes up toreach 25 power of scaling ratio for 120579 = 120∘ and 160∘ This iscaused by changing the length of high radiance section and

01

1

10

100

0 30 60 90 120 150 180

Phas

e fun

ctio

nΦ

Scattering angle 120579

120582 = 2120583m120582 = 3120583m120582 = 4120583m

120582 = 5120583m120582 = 6120583m

Figure 8 Calculated scattering phase function of Al2O3particles

with Mie model

radiance attenuation of the cold gases in tail section of theplume As plumersquos radiance mainly comes from the centralstripe which has a higher temperature and concentration forboth gases and Al

2O3particles Radiance attenuation of the

tail cold gases increases more rapidly than the projected areaon the surface of the plume with the growth of the scalingratio making growing rate of 119868119868

0being smaller than that of

surface area the latter has a growing rate of 1199032 in small zenithangle With the increase of zenith angle the effect of radianceattenuation decreases greatly and causes the ratio of radiationintensity increasing to reach and surpass 119903

2In the viewing of spectral characteristics for solid rocket

plume with different scaling ratios the variation of radiationintensity with wavelength for the 9 reduced scaling rocketmodels and the full scaled one in two directions 120579 =30∘ and 90∘ with 120593 = 30∘ has been plotted in Figure 10One can see good similarity between spectral variations ofdifferent scaling rocket models There are two peak spectraof radiation intensity 27sim30 120583mand 43sim46 120583m Comparedwith Figure 5 it is found that the former spectrum is thefeature radiative spectrum of H

2O the latter is of CO

2and

CO Besides in the two peak spectra spectral radiationintensity for plume of the 10 rockets decreases with theincreasing wavelength in 2sim6 120583mwaveband Since the Al

2O3

particle is the major radiation composition in the plumewhich has the maximum radiation for 120582 lt 20 120583m as particlethe temperature is higher than 2500K

The spectral radiation intensity of plume in all the zenithangles with 120593 = 30∘ for the 05 scaling rocket model and thefull scaled one has been shown in Figure 11 results of wave-length 120582 = 2 120583m 3 120583m and 4 120583m have been plotted With theincrease of the zenith angle radiation intensity of the plumefirstly increases arrives at a maximum value in a middleangle and decreases after that angle since the projected areaon plume surface changes with the zenith angle Comparingradiation intensity of plume in the front and rear hemisphereitrsquos found that is stronger in the rear hemisphere As read endsurface of plume has a strong radiance in the rear hemisphere

8 Journal of Applied Mathematics

0

03

06

09

12

15

0 02 04 06 08 1

II 0

120579 = 0∘ 120593 = 30∘

(a)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 40∘ 120593 = 30∘

(b)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 80∘ 120593 = 30∘

(c)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 120∘ 120593 = 30∘

(d)

0

02

04

06

08

1

0 02 04 06 08 1

120582 = 2120583m120582 = 3120583m120582 = 4120583m120582 = 5120583m

120582 = 2sim6120583mr15

r2

r25

II 0

120579 = 160∘ 120593 = 30∘

(e)

Figure 9 Ratio of plume radiation in the reduced scaling and full size rockets

Journal of Applied Mathematics 9

0

10000

20000

30000

40000

2 3 4 5 6

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

120582 (120583m)

I 120582W

(srmiddot120583

m)

(a)

0

10000

20000

30000

40000

I 120582W

(srmiddot120583

m)

2 3 4 5 6120582 (120583m)

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

(b)

Figure 10 Spectral radiation intensity of plume surface

0

2000

4000

6000

8000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

I 120582W

(srmiddot120583

m)

r = 05

(a)

0

10000

20000

30000

40000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

r = 1

I 120582W

(srmiddot120583

m)

(b)

Figure 11 Variation of radiation intensity of plume with zenith angles

Effect of radiance from the rear end also makes the directionwith maximum radiation intensity deflects toward the rearhemisphere which is remarkable for 119903 = 05 and themaximum radiation intensity arrives at 120579 = 50∘ in that caseThe colder section in the tail of plume is shorter and theradiance from the rear end surface would be bigger for asmaller scaling rocket

Figure 12 shows map of spectral radiance on 119909-119911 coordi-nate plane in plume flowof the 05 scaling solid rocket and thefull scaled ones for wavelength 120582 = 3 120583m and 5 120583m in 120579 = 0∘45∘ and 90∘ The radiance map of the two solid rockets hassimilar shape and close magnitude It is also found that the

high light area with a strong radiance is located in the centralstripe of the plume where temperature and concentrationof gases and Al

2O3particles are very high Compared the

width of the high light stripe in radiance maps of the twowavelengths it is narrower for 120582 = 3 120583m than 120582 = 5 120583mAs the radiation of Al

2O3particles plays the main role in

plume radiance of 3120583m the area with dense concentration ofparticles is smaller in radius than plume gases causing highradiance area centralizing in the plume in that wavelengthPlume gases are the major radiation composition for 120582 =5 120583m which has a bigger expansion angle and radius sizeand cause a wider stripe with high radiance Viewing the axial

10 Journal of Applied Mathematics

times102

z z z

z z z

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

r = 1 120582 = 3120583m 120579 = 0∘ r = 1 120582 = 3120583m 120579 = 45∘ r = 1 120582 = 3120583m 120579 = 90∘

r = 1 120582 = 5120583m 120579 = 0∘ r = 1 120582 = 5120583m 120579 = 45∘ r = 1 120582 = 5120583m 120579 = 90∘

1 2 3 4 5 6 7 8 9 10 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900times102

z

x

2 3 4 5 6 7 8 9 10 11 12

0

0 50 100 150 200 250 300

0 50 100 150 200 250 300 0 50 100 150 200 250 300

50 100 150 200 250 300 0

50

25

75

100

125

225

175

150

200

250

0 50 100 150 200 250

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

r = 05 120582 = 3120583m 120579 = 0∘ r = 05 120582 = 3120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

r = 05 120582 = 5120583m 120579 = 0∘ r = 05 120582 = 5120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

Figure 12 Map of spectral radiance on 119909-119911 coordinate plane

length of the high light stripe it is in the front part with halflength of the plume for the 05 scaling model but extendsfrom the front to end part of plume for the full scaled rocketmodel

8 Conclusion

Infrared radiation of plume from one full scaled solid rocketas well as the 01sim09 reducing scaled models in groundcondition has been investigated to study the similarity ofplumersquos infrared radiation for rocket models with differentscaling ratios Flowing field in the rocket and plume hasbeen computed with CFD code the omnidirectional andinfrared radiance on symmetric plane in the plume has beencalculated with the developed FVM codeThe research showsthat Al

2O3particle is the major radiation composition in the

rocket plumewhose scattering coefficient ismuch larger than

its absorption coefficient Ratio of radiation intensity fromreduced scaling plume to that of the full scaled one increaseswith the scaling ratio The power of the growth curve is 15for azimuth angle 120579 = 0∘ becomes 2 for 120579 = 40∘ and arrives at25 for 120579 = 120∘ and 160∘ There is good similarity betweenspectral variations of plumes from different scaling solidrockets and there are two peak spectra of radiation intensityin 27sim30 120583m and 43sim46 120583m Radiation intensity of theplume in rear hemisphere is bigger than that in the front andthe effect of radiance from the rear end surface makes thedirection with maximum radiation intensity deflects towardthe rear hemisphere

Conflict of Interests

There is not conflict of interests for all authors of this paper

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

0

2

4

6

8

10

12

2 3 4 5 6

All

120582 (120583m)

d = 4120583m d = 6120583md = 8120583m d = 10120583md = 14120583m d = 16120583m

120590(c

mminus

1 )

Figure 7 Calculated scattering coefficient of Al2O3particles with

Mie model

that in rear directions and scattering phase function instraight forward direction is the largest among all the fivewavelengths Comparing the results for the five wavelengthsin straight forward direction the smaller wavelength has alarger forward scattering phase function

7 Results and Discussion

The calculation method of spectral property of plume gasesand Al

2O3particles and infrared radiation intensity has been

verified with a reduced scaling solid rocket in a vacuumchamber Infrared radiation intensity in wavebands 27sim295 120583m and 42sim445 120583m at three zenith angles 120579 = 60∘90∘ and 120∘ have been calculated and compared with testresults The maximum difference between calculated and testresults is 209 which arrives at 120579 = 90∘ and the minimumdifference is 104 which arrives at 120579 = 60∘ [22]

Studying the similarities of plumersquos radiation intensityfrom solid rocket with different scale and ratio of plumeradiation between the reduced scaling and full size rockets inwavelengths 120582 = 2 120583m 3 120583m 4 120583m and 5 120583m and waveband2sim6120583m have been calculated and illustrated in Figure 9 Thehorizontal ordinate 119903 in the figure means the scaling ratioof rocket model and the vertical ordinate 119868119868

0means the

ratio of plumersquos radiation intensityThe three dashed lines arecorresponding to the 15 2 and 25 power function of scalingratio Results of 120579 = 0∘ 40∘ 80∘ 120∘ and 160∘ with 120593 = 30∘have been investigated It can be seen in the figure that ratioof radiation intensity increases with the increasing scalingratio for all the four wavelengths and waveband 2sim6 120583mTheincreasing rule of radiation intensity ratio for 120579 = 0∘ gets closeto the curve of 15 power mostly which gets closer to the 2-power function of scaling ratio for 120579 = 40∘ and comes up toreach 25 power of scaling ratio for 120579 = 120∘ and 160∘ This iscaused by changing the length of high radiance section and

01

1

10

100

0 30 60 90 120 150 180

Phas

e fun

ctio

nΦ

Scattering angle 120579

120582 = 2120583m120582 = 3120583m120582 = 4120583m

120582 = 5120583m120582 = 6120583m

Figure 8 Calculated scattering phase function of Al2O3particles

with Mie model

radiance attenuation of the cold gases in tail section of theplume As plumersquos radiance mainly comes from the centralstripe which has a higher temperature and concentration forboth gases and Al

2O3particles Radiance attenuation of the

tail cold gases increases more rapidly than the projected areaon the surface of the plume with the growth of the scalingratio making growing rate of 119868119868

0being smaller than that of

surface area the latter has a growing rate of 1199032 in small zenithangle With the increase of zenith angle the effect of radianceattenuation decreases greatly and causes the ratio of radiationintensity increasing to reach and surpass 119903

2In the viewing of spectral characteristics for solid rocket

plume with different scaling ratios the variation of radiationintensity with wavelength for the 9 reduced scaling rocketmodels and the full scaled one in two directions 120579 =30∘ and 90∘ with 120593 = 30∘ has been plotted in Figure 10One can see good similarity between spectral variations ofdifferent scaling rocket models There are two peak spectraof radiation intensity 27sim30 120583mand 43sim46 120583m Comparedwith Figure 5 it is found that the former spectrum is thefeature radiative spectrum of H

2O the latter is of CO

2and

CO Besides in the two peak spectra spectral radiationintensity for plume of the 10 rockets decreases with theincreasing wavelength in 2sim6 120583mwaveband Since the Al

2O3

particle is the major radiation composition in the plumewhich has the maximum radiation for 120582 lt 20 120583m as particlethe temperature is higher than 2500K

The spectral radiation intensity of plume in all the zenithangles with 120593 = 30∘ for the 05 scaling rocket model and thefull scaled one has been shown in Figure 11 results of wave-length 120582 = 2 120583m 3 120583m and 4 120583m have been plotted With theincrease of the zenith angle radiation intensity of the plumefirstly increases arrives at a maximum value in a middleangle and decreases after that angle since the projected areaon plume surface changes with the zenith angle Comparingradiation intensity of plume in the front and rear hemisphereitrsquos found that is stronger in the rear hemisphere As read endsurface of plume has a strong radiance in the rear hemisphere

8 Journal of Applied Mathematics

0

03

06

09

12

15

0 02 04 06 08 1

II 0

120579 = 0∘ 120593 = 30∘

(a)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 40∘ 120593 = 30∘

(b)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 80∘ 120593 = 30∘

(c)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 120∘ 120593 = 30∘

(d)

0

02

04

06

08

1

0 02 04 06 08 1

120582 = 2120583m120582 = 3120583m120582 = 4120583m120582 = 5120583m

120582 = 2sim6120583mr15

r2

r25

II 0

120579 = 160∘ 120593 = 30∘

(e)

Figure 9 Ratio of plume radiation in the reduced scaling and full size rockets

Journal of Applied Mathematics 9

0

10000

20000

30000

40000

2 3 4 5 6

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

120582 (120583m)

I 120582W

(srmiddot120583

m)

(a)

0

10000

20000

30000

40000

I 120582W

(srmiddot120583

m)

2 3 4 5 6120582 (120583m)

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

(b)

Figure 10 Spectral radiation intensity of plume surface

0

2000

4000

6000

8000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

I 120582W

(srmiddot120583

m)

r = 05

(a)

0

10000

20000

30000

40000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

r = 1

I 120582W

(srmiddot120583

m)

(b)

Figure 11 Variation of radiation intensity of plume with zenith angles

Effect of radiance from the rear end also makes the directionwith maximum radiation intensity deflects toward the rearhemisphere which is remarkable for 119903 = 05 and themaximum radiation intensity arrives at 120579 = 50∘ in that caseThe colder section in the tail of plume is shorter and theradiance from the rear end surface would be bigger for asmaller scaling rocket

Figure 12 shows map of spectral radiance on 119909-119911 coordi-nate plane in plume flowof the 05 scaling solid rocket and thefull scaled ones for wavelength 120582 = 3 120583m and 5 120583m in 120579 = 0∘45∘ and 90∘ The radiance map of the two solid rockets hassimilar shape and close magnitude It is also found that the

high light area with a strong radiance is located in the centralstripe of the plume where temperature and concentrationof gases and Al

2O3particles are very high Compared the

width of the high light stripe in radiance maps of the twowavelengths it is narrower for 120582 = 3 120583m than 120582 = 5 120583mAs the radiation of Al

2O3particles plays the main role in

plume radiance of 3120583m the area with dense concentration ofparticles is smaller in radius than plume gases causing highradiance area centralizing in the plume in that wavelengthPlume gases are the major radiation composition for 120582 =5 120583m which has a bigger expansion angle and radius sizeand cause a wider stripe with high radiance Viewing the axial

10 Journal of Applied Mathematics

times102

z z z

z z z

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

r = 1 120582 = 3120583m 120579 = 0∘ r = 1 120582 = 3120583m 120579 = 45∘ r = 1 120582 = 3120583m 120579 = 90∘

r = 1 120582 = 5120583m 120579 = 0∘ r = 1 120582 = 5120583m 120579 = 45∘ r = 1 120582 = 5120583m 120579 = 90∘

1 2 3 4 5 6 7 8 9 10 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900times102

z

x

2 3 4 5 6 7 8 9 10 11 12

0

0 50 100 150 200 250 300

0 50 100 150 200 250 300 0 50 100 150 200 250 300

50 100 150 200 250 300 0

50

25

75

100

125

225

175

150

200

250

0 50 100 150 200 250

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

r = 05 120582 = 3120583m 120579 = 0∘ r = 05 120582 = 3120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

r = 05 120582 = 5120583m 120579 = 0∘ r = 05 120582 = 5120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

Figure 12 Map of spectral radiance on 119909-119911 coordinate plane

length of the high light stripe it is in the front part with halflength of the plume for the 05 scaling model but extendsfrom the front to end part of plume for the full scaled rocketmodel

8 Conclusion

Infrared radiation of plume from one full scaled solid rocketas well as the 01sim09 reducing scaled models in groundcondition has been investigated to study the similarity ofplumersquos infrared radiation for rocket models with differentscaling ratios Flowing field in the rocket and plume hasbeen computed with CFD code the omnidirectional andinfrared radiance on symmetric plane in the plume has beencalculated with the developed FVM codeThe research showsthat Al

2O3particle is the major radiation composition in the

rocket plumewhose scattering coefficient ismuch larger than

its absorption coefficient Ratio of radiation intensity fromreduced scaling plume to that of the full scaled one increaseswith the scaling ratio The power of the growth curve is 15for azimuth angle 120579 = 0∘ becomes 2 for 120579 = 40∘ and arrives at25 for 120579 = 120∘ and 160∘ There is good similarity betweenspectral variations of plumes from different scaling solidrockets and there are two peak spectra of radiation intensityin 27sim30 120583m and 43sim46 120583m Radiation intensity of theplume in rear hemisphere is bigger than that in the front andthe effect of radiance from the rear end surface makes thedirection with maximum radiation intensity deflects towardthe rear hemisphere

Conflict of Interests

There is not conflict of interests for all authors of this paper

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

0

03

06

09

12

15

0 02 04 06 08 1

II 0

120579 = 0∘ 120593 = 30∘

(a)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 40∘ 120593 = 30∘

(b)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 80∘ 120593 = 30∘

(c)

0

02

04

06

08

1

0 02 04 06 08 1

II 0

120579 = 120∘ 120593 = 30∘

(d)

0

02

04

06

08

1

0 02 04 06 08 1

120582 = 2120583m120582 = 3120583m120582 = 4120583m120582 = 5120583m

120582 = 2sim6120583mr15

r2

r25

II 0

120579 = 160∘ 120593 = 30∘

(e)

Figure 9 Ratio of plume radiation in the reduced scaling and full size rockets

Journal of Applied Mathematics 9

0

10000

20000

30000

40000

2 3 4 5 6

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

120582 (120583m)

I 120582W

(srmiddot120583

m)

(a)

0

10000

20000

30000

40000

I 120582W

(srmiddot120583

m)

2 3 4 5 6120582 (120583m)

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

(b)

Figure 10 Spectral radiation intensity of plume surface

0

2000

4000

6000

8000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

I 120582W

(srmiddot120583

m)

r = 05

(a)

0

10000

20000

30000

40000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

r = 1

I 120582W

(srmiddot120583

m)

(b)

Figure 11 Variation of radiation intensity of plume with zenith angles

Effect of radiance from the rear end also makes the directionwith maximum radiation intensity deflects toward the rearhemisphere which is remarkable for 119903 = 05 and themaximum radiation intensity arrives at 120579 = 50∘ in that caseThe colder section in the tail of plume is shorter and theradiance from the rear end surface would be bigger for asmaller scaling rocket

Figure 12 shows map of spectral radiance on 119909-119911 coordi-nate plane in plume flowof the 05 scaling solid rocket and thefull scaled ones for wavelength 120582 = 3 120583m and 5 120583m in 120579 = 0∘45∘ and 90∘ The radiance map of the two solid rockets hassimilar shape and close magnitude It is also found that the

high light area with a strong radiance is located in the centralstripe of the plume where temperature and concentrationof gases and Al

2O3particles are very high Compared the

width of the high light stripe in radiance maps of the twowavelengths it is narrower for 120582 = 3 120583m than 120582 = 5 120583mAs the radiation of Al

2O3particles plays the main role in

plume radiance of 3120583m the area with dense concentration ofparticles is smaller in radius than plume gases causing highradiance area centralizing in the plume in that wavelengthPlume gases are the major radiation composition for 120582 =5 120583m which has a bigger expansion angle and radius sizeand cause a wider stripe with high radiance Viewing the axial

10 Journal of Applied Mathematics

times102

z z z

z z z

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

r = 1 120582 = 3120583m 120579 = 0∘ r = 1 120582 = 3120583m 120579 = 45∘ r = 1 120582 = 3120583m 120579 = 90∘

r = 1 120582 = 5120583m 120579 = 0∘ r = 1 120582 = 5120583m 120579 = 45∘ r = 1 120582 = 5120583m 120579 = 90∘

1 2 3 4 5 6 7 8 9 10 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900times102

z

x

2 3 4 5 6 7 8 9 10 11 12

0

0 50 100 150 200 250 300

0 50 100 150 200 250 300 0 50 100 150 200 250 300

50 100 150 200 250 300 0

50

25

75

100

125

225

175

150

200

250

0 50 100 150 200 250

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

r = 05 120582 = 3120583m 120579 = 0∘ r = 05 120582 = 3120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

r = 05 120582 = 5120583m 120579 = 0∘ r = 05 120582 = 5120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

Figure 12 Map of spectral radiance on 119909-119911 coordinate plane

length of the high light stripe it is in the front part with halflength of the plume for the 05 scaling model but extendsfrom the front to end part of plume for the full scaled rocketmodel

8 Conclusion

Infrared radiation of plume from one full scaled solid rocketas well as the 01sim09 reducing scaled models in groundcondition has been investigated to study the similarity ofplumersquos infrared radiation for rocket models with differentscaling ratios Flowing field in the rocket and plume hasbeen computed with CFD code the omnidirectional andinfrared radiance on symmetric plane in the plume has beencalculated with the developed FVM codeThe research showsthat Al

2O3particle is the major radiation composition in the

rocket plumewhose scattering coefficient ismuch larger than

its absorption coefficient Ratio of radiation intensity fromreduced scaling plume to that of the full scaled one increaseswith the scaling ratio The power of the growth curve is 15for azimuth angle 120579 = 0∘ becomes 2 for 120579 = 40∘ and arrives at25 for 120579 = 120∘ and 160∘ There is good similarity betweenspectral variations of plumes from different scaling solidrockets and there are two peak spectra of radiation intensityin 27sim30 120583m and 43sim46 120583m Radiation intensity of theplume in rear hemisphere is bigger than that in the front andthe effect of radiance from the rear end surface makes thedirection with maximum radiation intensity deflects towardthe rear hemisphere

Conflict of Interests

There is not conflict of interests for all authors of this paper

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

0

10000

20000

30000

40000

2 3 4 5 6

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

120582 (120583m)

I 120582W

(srmiddot120583

m)

(a)

0

10000

20000

30000

40000

I 120582W

(srmiddot120583

m)

2 3 4 5 6120582 (120583m)

r = 01

r = 02

r = 03

r = 04r = 05

r = 06

r = 07r = 08

r = 09

r = 1

(b)

Figure 10 Spectral radiation intensity of plume surface

0

2000

4000

6000

8000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

I 120582W

(srmiddot120583

m)

r = 05

(a)

0

10000

20000

30000

40000

0 50 100 150 200120579 (∘)

120582 = 2120583m120582 = 3120583m120582 = 4120583m

r = 1

I 120582W

(srmiddot120583

m)

(b)

Figure 11 Variation of radiation intensity of plume with zenith angles

Effect of radiance from the rear end also makes the directionwith maximum radiation intensity deflects toward the rearhemisphere which is remarkable for 119903 = 05 and themaximum radiation intensity arrives at 120579 = 50∘ in that caseThe colder section in the tail of plume is shorter and theradiance from the rear end surface would be bigger for asmaller scaling rocket

Figure 12 shows map of spectral radiance on 119909-119911 coordi-nate plane in plume flowof the 05 scaling solid rocket and thefull scaled ones for wavelength 120582 = 3 120583m and 5 120583m in 120579 = 0∘45∘ and 90∘ The radiance map of the two solid rockets hassimilar shape and close magnitude It is also found that the

high light area with a strong radiance is located in the centralstripe of the plume where temperature and concentrationof gases and Al

2O3particles are very high Compared the

width of the high light stripe in radiance maps of the twowavelengths it is narrower for 120582 = 3 120583m than 120582 = 5 120583mAs the radiation of Al

2O3particles plays the main role in

plume radiance of 3120583m the area with dense concentration ofparticles is smaller in radius than plume gases causing highradiance area centralizing in the plume in that wavelengthPlume gases are the major radiation composition for 120582 =5 120583m which has a bigger expansion angle and radius sizeand cause a wider stripe with high radiance Viewing the axial

10 Journal of Applied Mathematics

times102

z z z

z z z

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

r = 1 120582 = 3120583m 120579 = 0∘ r = 1 120582 = 3120583m 120579 = 45∘ r = 1 120582 = 3120583m 120579 = 90∘

r = 1 120582 = 5120583m 120579 = 0∘ r = 1 120582 = 5120583m 120579 = 45∘ r = 1 120582 = 5120583m 120579 = 90∘

1 2 3 4 5 6 7 8 9 10 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900times102

z

x

2 3 4 5 6 7 8 9 10 11 12

0

0 50 100 150 200 250 300

0 50 100 150 200 250 300 0 50 100 150 200 250 300

50 100 150 200 250 300 0

50

25

75

100

125

225

175

150

200

250

0 50 100 150 200 250

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

r = 05 120582 = 3120583m 120579 = 0∘ r = 05 120582 = 3120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

r = 05 120582 = 5120583m 120579 = 0∘ r = 05 120582 = 5120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

Figure 12 Map of spectral radiance on 119909-119911 coordinate plane

length of the high light stripe it is in the front part with halflength of the plume for the 05 scaling model but extendsfrom the front to end part of plume for the full scaled rocketmodel

8 Conclusion

Infrared radiation of plume from one full scaled solid rocketas well as the 01sim09 reducing scaled models in groundcondition has been investigated to study the similarity ofplumersquos infrared radiation for rocket models with differentscaling ratios Flowing field in the rocket and plume hasbeen computed with CFD code the omnidirectional andinfrared radiance on symmetric plane in the plume has beencalculated with the developed FVM codeThe research showsthat Al

2O3particle is the major radiation composition in the

rocket plumewhose scattering coefficient ismuch larger than

its absorption coefficient Ratio of radiation intensity fromreduced scaling plume to that of the full scaled one increaseswith the scaling ratio The power of the growth curve is 15for azimuth angle 120579 = 0∘ becomes 2 for 120579 = 40∘ and arrives at25 for 120579 = 120∘ and 160∘ There is good similarity betweenspectral variations of plumes from different scaling solidrockets and there are two peak spectra of radiation intensityin 27sim30 120583m and 43sim46 120583m Radiation intensity of theplume in rear hemisphere is bigger than that in the front andthe effect of radiance from the rear end surface makes thedirection with maximum radiation intensity deflects towardthe rear hemisphere

Conflict of Interests

There is not conflict of interests for all authors of this paper

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

times102

z z z

z z z

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

2

0

minus20 5 10 15 20

x

r = 1 120582 = 3120583m 120579 = 0∘ r = 1 120582 = 3120583m 120579 = 45∘ r = 1 120582 = 3120583m 120579 = 90∘

r = 1 120582 = 5120583m 120579 = 0∘ r = 1 120582 = 5120583m 120579 = 45∘ r = 1 120582 = 5120583m 120579 = 90∘

1 2 3 4 5 6 7 8 9 10 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800900times102

z

x

2 3 4 5 6 7 8 9 10 11 12

0

0 50 100 150 200 250 300

0 50 100 150 200 250 300 0 50 100 150 200 250 300

50 100 150 200 250 300 0

50

25

75

100

125

225

175

150

200

250

0 50 100 150 200 250

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

z

x

1

0

0 2 4 6 8 10

r = 05 120582 = 3120583m 120579 = 0∘ r = 05 120582 = 3120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

r = 05 120582 = 5120583m 120579 = 0∘ r = 05 120582 = 5120583m 120579 = 45∘ r = 05 120582 = 3120583m 120579 = 90∘

Figure 12 Map of spectral radiance on 119909-119911 coordinate plane

length of the high light stripe it is in the front part with halflength of the plume for the 05 scaling model but extendsfrom the front to end part of plume for the full scaled rocketmodel

8 Conclusion

Infrared radiation of plume from one full scaled solid rocketas well as the 01sim09 reducing scaled models in groundcondition has been investigated to study the similarity ofplumersquos infrared radiation for rocket models with differentscaling ratios Flowing field in the rocket and plume hasbeen computed with CFD code the omnidirectional andinfrared radiance on symmetric plane in the plume has beencalculated with the developed FVM codeThe research showsthat Al

2O3particle is the major radiation composition in the

rocket plumewhose scattering coefficient ismuch larger than

its absorption coefficient Ratio of radiation intensity fromreduced scaling plume to that of the full scaled one increaseswith the scaling ratio The power of the growth curve is 15for azimuth angle 120579 = 0∘ becomes 2 for 120579 = 40∘ and arrives at25 for 120579 = 120∘ and 160∘ There is good similarity betweenspectral variations of plumes from different scaling solidrockets and there are two peak spectra of radiation intensityin 27sim30 120583m and 43sim46 120583m Radiation intensity of theplume in rear hemisphere is bigger than that in the front andthe effect of radiance from the rear end surface makes thedirection with maximum radiation intensity deflects towardthe rear hemisphere

Conflict of Interests

There is not conflict of interests for all authors of this paper

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

Acknowledgments

This work is cosponsored by the National Natural ScienceFoundation of China (51376065 51176052) Guangdong KeyScientific Project (2013B010405004) Guangdong ProvinceKey Laboratory of Efficient and Clean Energy Utilization(2013A061401005) South China University of Technologyand Key Laboratory of Efficient and Clean Energy Utilizationof Guangdong Higher Education Institutes (KLB10004)

References

[1] G L Wang ldquoDesign of solid rocket enginerdquo chapter 3 Publish-ing House of Northwest University Xirsquoan China 1994

[2] V V Rozanov and A I Lyapustin ldquoSimilarity of radiativetransfer equation error analysis of phase function truncationtechniquesrdquo Journal of Quantitative Spectroscopy and RadiativeTransfer vol 111 no 12-13 pp 1964ndash1979 2010

[3] Z Jun and P Wenbing ldquoSimilarity transformations of radiationhydrodynamic equations and investigation on laws of radiativeconductionrdquo Physics of Fluids B vol 4 no 4 pp 872ndash876 1992

[4] T Duracz and N J Mccormick ldquoEquations for estimatingthe similarity parameter from radiation measurements withinweakly absorbing optically thick cloudsrdquo Journal of the Atmo-spheric Sciences vol 43 no 5 pp 486ndash492 1986

[5] CMitrescu andG L Stephens ldquoOn similarity and scaling of theradiative transfer equationrdquo Journal ofQuantitative Spectroscopyand Radiative Transfer vol 86 no 4 pp 387ndash394 2004

[6] A I Bril V P Kabashnikov N V Kuzmina and V M PopovldquoSimilarity of heat radiation from turbulent buoyant jetsrdquoInternational Journal of Heat and Mass Transfer vol 41 no 10pp 1347ndash1356 1998

[7] D K Edwards ldquoMolecular gas band radiationrdquo Advances inHeat Transfer vol 12 pp 115ndash193 1976

[8] WMalkmus ldquoRandom Lorentz bandmodel with exponentiallytailed Sminus1 line intensity distribution functionrdquo Journal of theOptical Society of America vol 57 no 3 p 323 1967

[9] JMHartmann R Levi Di Leon and J Taine ldquoLine-by-line andnarrow-band statistical model calculations for H

2Ordquo Journal of

Quantitative Spectroscopy and Radiative Transfer vol 32 no 2pp 119ndash127 1984

[10] M F Modest ldquoWeighted-sum-of-gray-gases model for arbi-trary solution methods in radiative transferrdquo Journal of HeatTransfer vol 113 no 3 pp 650ndash656 1991

[11] R A Reed and V S Calia ldquoReview of aluminum oxiderocket exhaust particlesrdquo in Proceedings of the 28th AIAAThermophysics Conference AIAA-93-2819 Orlando Fla USAJuly 1993

[12] R A Reed and V S Calia ldquoNew measurements of liquidaluminum oxiderdquo in Proceedings of the JANNAF Exhaust PlumeSubcommittee Meeting Philips Laboratory Kirland AFB Albu-querque NM USA 1993

[13] C F Bohran and D R Huffman Absorption and Scattering ofLight by Small Particles JohnWileyampSonsNewYorkNYUSA1983

[14] Q Yu L Liu Y Pan D Zhang J Ji and H Tan ldquoMonteCarlo method for simulating the radiative characteristics of ananisotropic mediumrdquoHeat Transfer Asian Research vol 28 no3 pp 201ndash210 1999

[15] D A Blank and S C Mishra ldquoUse of the 2-D collapseddimensionmethod in gray enclosures with absorbing-emitting-isotropic scattering media in radiative equilibriumrdquo NumericalHeat Transfer Part B vol 30 no 4 pp 469ndash481 1996

[16] N Selcuk and N Kayakol ldquoEvaluation of discrete ordinalesmethod for radiative transfer in rectangular furnacesrdquo Interna-tional Journal of Heat and Mass Transfer vol 40 no 2 pp 213ndash222 1997

[17] E H Chui G D Raithby and P M J Hughes ldquoPredictionof radiative transfer in cylindrical enclosures with the finitevolume methodrdquo Journal of Thermophysics and Heat Transfervol 6 no 4 pp 605ndash611 1992

[18] Z Li H-J Xiang and X-Y Zhang ldquoNumerical simulationof composite solid propellant rocket motor exhaust plumerdquoJournal of Solid Rocket Technology vol 37 no 1 pp 37ndash42 2014

[19] P C Braithwaite ldquoQuench bomb investigation of aluminumoxide formation from solid rocket propellants (part I) exper-imental methodologyrdquo in Proceedings of the 25th JANNAFCombustion Meeting CPIA Publication 498 1988

[20] S-W Fan X-Y Zhang D-Q Zhu H-J Xiang and G-B CaildquoCalculation of the infrared characteristics of the solid rocketplume with FVM methodrdquo Journal of Astronautics vol 26 no6 pp 793ndash797 2005

[21] C B Ludwig W Malkmus J E Reardon et al Handbookof Infrared Radiation from Combustion Gases NASA-SP-3080Hamppon Virainiaz Lanzley Research Center 1973

[22] D Q Zhu X Y Zhang H J Xiang et al ldquoCooling techniquesfor high pressure chamber of LOXkerosene enginerdquo Journal ofAstronautics vol 29 no 1 pp 255ndash259 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of