74

CHAPTER 5

ROSIN-RAMMLER DISTRIBUTION AND

DEFINITION OF THE

COMPUTATIONAL DOMAIN

5.1. BACKGROUND

Exact analysis of radiative heating estimation from the exhaust plume of solid

rockets requires two parts, particle flow field analysis and radiation analysis. Both are very

complicated and require tremendous efforts, as shown by Girata Jr. and McGregor [1984],

and hence techniques to predict plume radiation analysis without flow field analysis have

been proposed from the macroscopic view point. In this study, the computational domain is

defined as three dimensional volume elements with exhaust plume with a computed

temperature and pressure profiles by the CFD code. Aluminium Oxide solid particles

present in the exhaust plume of Solid Rocket Motor (SRM) significantly influences the

radiosity of exhaust plume of solid motors. Further, particle spectrum details become an

input for predicting the specific impulse of SRM. Emission and scattering characteristics of

the exhaust plume are depending on the particle characteristics namely size, mass, velocity

and temperature, which are not adequately defined, in the published literature. There are

several mathematical functions describing particle size distributions. The most popular

among them is Rosin-Rammler distribution which uses a semi-empirical technique to

describe the particle distribution using only two parameters. The origin and application fields

of Rosin-Rammler distributions are well explained by Wilbur Brown and Kenneth Wohletz

[1995]. Edwards and Bobco [1982], Edwards and Babikian [1990] have used Rosin-

Rammler distributions in their analysis for plume radiosity as the model for particle

spectrum. Yasuhi Sakurai and Hiroshi Kimura [1986] reviewed the topic of plume radiation

from a solid rocket and considered uniform percentage (20%) of alumina for five different

particle sizes in the analysis.

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This study makes use of particle spectrum with their respective mass portions in the

exhaust plume of SRM by taking Rosin-Rammler distribution as its basis. Thus the particle

spectrum data given by this study becomes the input for predicting the specific impulse of

SRM and generating more accurate scattering cross sections present in the Radiative

Transfer Equation (RTE).

5.2. INTRODUCTION

Propellants are the materials generating a large number of gaseous molecules at

high temperature during combustion and which can self-sustain without the presence of

ambient oxidizer combustion. A rocket motor is a typical example of an energy transfer

system, which can be directly explained by thermodynamics and Newton’s second law. A

pressurized high temperature gas generated in the system is expanded adiabatically, and

the sensible energy of the gas is converted to kinetic energy. Thus the system produces a

reaction force. Hence the thermodynamic requirement for a rocket motor is to get a high

pressure and high temperature gas in the motor, Kubota [1984]. Advanced solid propellant

formulations contain metals to enhance the thrust of Solid Rocket Motors and to reduce the

combustion instability. The use of aluminium in the solid propellants is a widely accepted

technique Aluminium and its oxide components increase the flame temperature and reduce

the combustion instability. It has long been recognized that combustion instabilities in solid

rocket motors can be alleviated by adding powdered metals to the fuel mix. Therefore there

is a considerable body of literature and data concerning metal oxide particle produced by

solid propellant rocket motors. Theoretical performance of these propellants is generally

superior to that of non-metallized formulations. However, the predicted increase of specific

impulse efficiency of metallic formulations is reduced by the presence of metallic oxide

particles due to the lack of inter phase thermal and momentum equilibrium between the

oxide particles and the gas phase and the increased heat losses to the rocket hardware

resulting from radiation by the particles. The properties of the Al2O3 particles namely, size,

density, temperature, physical state and velocity play an ever increasing role in motor

performance, nozzle design, base heat transfer and infrared signature. Thus from the

engineering considerations, it is desirable to have particle spectrum details in the rocket

exhaust to predict the actual performance of the rocket motor.

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The Al2O3 particles produced by the solid rockets are not considered

hazardous in terms of their toxicity. However, they may play a role in

weather modification by seeding clouds, especially when they remain

airborne for extended time periods. There are three mechanisms by which

these particles can be removed from the atmosphere:

(1) Natural settlement on the ground due to gravitational force

(2) They can be washed out of the air by an overriding rainfall

(3) They become the nuclei of raindrops by condensation.

The effectiveness of each of these mechanisms is a strong function of the particle

size and hence particle spectrum needs to be explored. Aluminium oxide is produced as an

equilibrium condensed phase product of the combustion of solid propellants loaded with a

substantial fraction (10-20% by weight) of metallic powdered aluminium. As the liquid

particles pass through the nozzle into the plume, the gas temperature drops rapidly when

compared to that of particles, with the temperature lag depending on the particle size. The

particles also will solidify at a size dependent rate. The mass fraction of alumina in the

plume is predicted by standard rocket performance codes like NASA SP273 as referred by

Frederick Simmons [2000]. According to Dawbarn, Kinslow and Watson [1980], the Al2O3

particles collected and examined during various tests show that the dust particles ranging

from 0.01 to 0.1 µ are irregularly shaped and in many cases have agglomerated into

clusters. Further particles of size ranging from 1 to 4.5µ are approximately spherical shapes

because of surface tension and vapor pressure of the exhaust plume. The bigger particles

concentrate to the central axis of the plume while the smaller particles are accelerated more

rapidly in the radial direction by pressure gradients in the expanding plume. Radiosity of

plume largely depends on absorption, emission and scattering of radiant energy and the

Al2O3 particles contribute to these phenomena. Scattering efficiency of the particle cloud

depends on the wavelength of incident radiant energy and size of particle and hence the

modeling of scattering of radiant energy requires particle spectrum details of each control

volume defined in the computational domain of exhaust plume.

Studies on plume radiosity conducted by Morizumi and Carpenter [1964]

considered uniform mass fractions for each category of particles. However, the spatial

distribution of the solid particles in the plume volume is not uniform and Edwards and Bobco

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[1982] studied the effect of varying particle size distribution on plume radiosity by making

use of Bobco’s [1966] engineering model. This model basically makes use of an effective

emissivity of the particle cloud and an inverse power law for the plume radiosity. Later, the

temperature profile of the exhaust plume computed by CFD code clearly indicates that the

temperature is not steadily decreasing from the nozzle exit plane, but in between it is

increasing due to the series of shocks getting developed in the plume. Thus more realistic

plume radiosity is to be arrived at and which can be done only through the predicted

temperature profile by CFD.

Most studies of the size distribution points towards a bimodal size distribution for

the alumina in the exhaust plume of solid rockets. According to Brewaster [1989] a

monomodal representation of particle spectrum would be adequate if one mode dominates

the optical properties. This study makes use of the proportions of mass of each category of

particles as given by the Rosin-Rammler distributions. Then these mass fractions are

mapped into volumes of particles of different sizes predicted by their limiting trajectories

using CFD analysis. Modeling details of these two sections are addressed in this chapter.

5.3. THE ROSIN-RAMMLER DISTRIBUTION

The Rosin-Rammler distribution predicts the mass fraction w of particles having

size greater than the diameter D as

0

( )Nr

D

Dw e

−

= (5.1)

where the exponent Nr affects the spread of the distribution and D0 is a parameter affecting

the mean particle size of distribution. Thus this is a bimodal cumulative frequency

distribution. Figure 5.1 shows both the frequency and cumulative frequency of mass of

particles with parameters Do=1.0µ and Nr=1.5

The major parameters influencing the particle size are chamber pressure, throat

diameter and resident time inside the chamber and these are changing from motor to motor.

The particle distribution in each case can be obtained by fine tuning the bimodal parameters

D0 and Nr in the Rosin-Rammler distribution. The sensitivity of the two parameters D0 and Nr

in the Rosin-Rammler distribution is shown in Figures 5.2 and 5.3. Figure 5.2, shows that

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higher value of Nr give a tight spread for the distribution. The average particle size increases

as D0 increases. Figure 5.3 shows that the average particle size increases as D0 increases

0

0.2

0.4

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Cumulative fractionFraction : Case N

r = 5 , D

0 = 2 µµµµm

Particle diameter, µµµµm

Fra

cti

on

Figure 5.1: Rosin-Rammler distribution for the parameters

D0=2 µm and Nr =5

0

0.2

0.4

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

Cumulative fraction : D0 = 2 µµµµm ,Nr=5

Cumulative fraction : D0

= 3 µµµµm, Nr = 5

Fraction : Case D0

= 2 µµµµm , Nr = 5

Fraction : Case D0

= 3 µµµµm , Nr = 5

Particle diameter, µµµµm

Fra

cti

on

Figure 5.2 Rosin-Rammler distribution for the parameters

D0=2,3 µm and Nr =5

79

0

0.2

0.4

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Nr =5N

r = 4

Nr = 3

Nr = 2

Particle diameter, µµµµm

Fra

cti

on

D0=2 µµµµm

Fig.5.3 Comparison of particle distribution for constant mean particle size

0

0.2

0.4

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

D0 = 2 µµµµm

D0

= 3 µµµµmD

0 = 4 µµµµm

D0 = 5 µm

Particle diameter, µµµµm

Fra

cti

on

s

Nr=3

Fig.5.4 Comparison of particle distribution for different Do and for constant Nr

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5.4. CHARACTERISTICS OF SRM PLUME

The mathematical model for thermal radiation should describe radiation from Al2O3

particles contained in the gaseous products of combustion as well as emission from the

principal emitting gases, namely CO.CO2, H20 and HCl. Particles are accelerated in the

nozzle plume flow field due to the velocity gradients produced by pressure gradients exerted

on them by the gaseous flow. As the vehicle reaches higher altitudes, the atmospheric

pressure gets reduced much below the nozzle exit pressure and hence the plume becomes

under- expanded. At this stage, shape of the plume can be approximated as a truncated

cone emanating from the nozzle exit plane with a certain semi vertical angle, which

corresponds to the expansion angle of the plume. Consistent flight data of base heating

due to the exhaust plume of solid rockets show that irrespective of the altitude in the ascend

flight of a launch vehicle, the radiative component is almost constant till the regime where

the chamber pressure drastically reduces to almost 50 %. This indicates that the component

of radiation due to the gaseous particles is negligibly small and the change of shape of

gaseous boundary is insignificant for the base heating due to radiation. Further, a recent

analysis carried out by Jeya Bharata Reddy [2008] of particle trajectories shows the

following features which are derived from figs. 5.5 to 5.13;

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(1) Smaller particles are uniformly spread out in the plume as shown in Figures 5.5

& 5.6.

(2) The bigger particles, say above 4.5µ, are confined to the region of central core

as seen in Fig.5.13.

(3) The limiting trajectories of bigger particles are almost conical and each of their

domains contains particles smaller than them.

5.5. COMPUTATIONAL DOMAIN

The domain of computation for the Radiative Transfer Equation comprises of

volumetric cells representing the above discussed characteristics. The Rosin-Rammler

distribution gives the fraction of solid particles above a certain diameter. Since the

82

computational domain is volume, a mapping of mass of solid particles to volume is required

to model Rosin-Rammler distribution in the exhaust plume. Here in this study, the envelopes

of trajectories of particles of different particle sizes generated by CFD studies [Jeya Bharata

Reddy, 2008] are utilized for the mapping.

A two phase finite volume Navier-Stokes solver using an Euler-Lagrangean method

is employed to trace the particle trajectory. Here, the Eulerian method is applied for the gas

species while Lagrangean method is applied on particles. A one way coupling is

incorporated in the model where the gas temperature influences the particle temperature.

Two way coupling is not taken into account. From various radial locations at the inlet region

of the nozzle, Al2O3 particle of various sizes ranging from 0.5 to 10 µm is released (one type

of particle at a time) and its path is traced using a Lagrangean method.

The equation of particle trajectories of different particle sizes is generated using the

WINDIG software. Appendix 5.B provides the equations of particle plume boundary of

different particle sizes in the form y=f(x) where x is the axial length of trajectory from the

nozzle exit plane of the motor.

0

2

4

6

0 2.5 5.0 7.5

R=4.0 µµµµR-3.5 µµµµR=3.0 µµµµR=2.5 µµµµR=2.0 µµµµR=1.5 µµµµR=1.0 µµµµR=0.5 µµµµR=0.1 µµµµ

Length of Plume Beyond the Exit Plane, m

Ra

diu

s o

f th

e E

nv

elo

p,m

Fig.5.14 Envelops of particle boundaries of different sizes

83

The volume of integration of the curve y=f(x) when rotated along the X-axis

between the limits

hx ≤≤0 is

∫∏=h

dxyV

0

2

(5.2)

In general, a plume length of 6 to 7 nozzle exit diameters is found to be appropriate

for the convergence of radiative flux. In eq (5.2), the lower limit zero is assigned to the

station of the nozzle exit plane. The upper limit of the above integration, h, should be

assigned a value as length of plume from the nozzle exit plane satisfying the convergence

criterion. Figure 5.15 provides the values of partial volumes of particles of different radius as

a function of total volume. This data is generated for a solid motor with a nozzle exit

diameter of 3140 mm. It can be seen that the volume of particles with radius less than

0.5µm radius is exponentially increasing. In the case of modeling the plume radiosity using

Monte Carlo Techniques, one has to minimize the computational domain for a reasonably

faster solution. Hence the inclusion of tiny particles gives rise to large computational

domain, which in turn leads to very large computational time. However, exclusion of very

small particles in the analysis can be justified since they are less significant contributors to

the radiosity due to their lower temperature and emissivity.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5

AXIAL DISTANCE

0

25

50

75

100

125

150

175

200

225

250

275

300

VO

LU

ME

OF

PLU

ME

Meu

0.1

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Fig.5.15: Partial Volumes of different particles as a function of axial distance.

84

To enable the computation of the parameter h, the analytical expressions of the

integrand ∫ dxy2

applicable to different particle sizes up to 4.5µm is given in Appendix-5.C.

5.6. GEOMETRY OF COMPUTATIONAL DOMAIN

The computational domain of plume can be approximated in the axi-symmetric flow

field as concentric, as suggested by Watsomn and Lee [1976]. These conical surfaces are

emanating from a common vertex which is bounded along the axial coordinate by planes at

right angles to the conical surface axes. The boundary of particle trajectories of each type of

solid particles is considered as a cone with certain expansion angle. Geometrical details of

a typical conical surface containing ,say, ith category of particles is shown in Fig.5.16.

Fig.5.16: Sub volume Vi and its geometric parameters

Using eqn.(5.2) the partial volume of the ith type of particle is estimated as

∫∏=h

dxxyV ii

0

)(2

(5.3)

Integrating the RTE over the computational domain, one can estimate the radiosity

of the plume. For integrating the domain, the following three geometrical parameters are

needed for grid generation in the computational domain for a given Vi as predicted by

eq.(5.3) :

(1) the expansion angle of each category of particles, which is the semi vertical

angle of conical section confined to that category of particles, say dΦi ,as

shown in Fig.5.16

(2) radius of the conical shell at the nozzle exit plane, r1i, and

(3) radius of the conical at axial height of h, r2i. For a given volume of Vi the

expansion angle can be derived as

dΦi a

h

Nozzle

Exit Plane

85

])[(

3tan

33aha

Vd i

i−+

=π

ϕ

(5.4)

Then the radii r1i and r2i are given by

ii dar ϕtan1 = (5.5)

ii dhar ϕtan)(2 += (5.6)

Here ‘a’ is motor specific and hence can be used as an input.

0

1

2

3

20 40 60 80 100

Radius at Axial Height hRadius at the Nozzle Exit Plane

Computational Volume,m3

Rad

ii,m

Figure 5.17 Variations of Radii in Plume Geometry with Volume

Figure 5.17 shows the loci of r1i and r2i as a function of partial volume, Vi. This figure

reveals that the dimensional increase of the two radii at the top and bottom of the

computational domain of a plume is almost linear even when the volume is enhanced up to

five times. The loci of the radii are given below:

Equation of radius at the axial height, h, as a function of volume, Vi, is

2

2 ( ) 8.04 05 0.0267 0.626i i ir V E V V= − − + + (5.7)

And the equation of radius at the nozzle exit plane is

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2

1( ) 3.21 05 0.0108 0.262i i ir V E V V= − − + + (5.8)

5.7. PARTICLE DENSITY

Modeling of scattering of thermal radiant energy emanating from each solid particle

in the exhaust plume requires its position in the plume. However, such a task being

extremely complex and time consuming, one takes into account of solid particles contained

in a control volume and hence requires an estimate of particle density in each control

volume. The scattering cross section of each control volume thus depends on its particle

density.

A reasonable estimate of the number of particles in each sub volume, Vi, can be

made using the following equation.

∑=

=m

i

iiVNi1

ρ (5.9)

Where m is the number of different category of particles and ρi is the particle

number density of ith particle size present in the volume cell Vi. The particle number density

ρi in each cell can be estimated using the volume of ith cell and the mass flow rate of plume.

By making use of isentropic relations the mass flow rate of exhaust can be estimated as

follows.

1. 0

0

2 1

1 2p

Pw A

RT

γ

γ γ γ

γ

− +=

+ (5.10)

Volume of exhaust plume, Vp, exiting through the nozzle exit plane per second is

VL*A .where VL is exit velocity of plume and A is the nozzle exit area. Hence, numerically,

the mass of plume in Vp, is W.p Therefore, mass per unit volume of plume is

AV

W

L

p

*

.

and

this scaling ratio can be applied to estimate the mass of plume for any given volume. Total

mass of particles in any volume, V, is therefore

f

L

p

p wAV

wVm *

**

.

= (5.11)

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where Wf is the fraction of mass of solid particles for a given loading ratio of aluminium in

the solid propellant. Now, the mass fractions of each category of particles in the volume, V,

can be related to Rosin-Rammler distribution as

∑=

=m

i

ip mm1

(5.12)

where

−=

−−

f

L

pD

D

D

D

i wAV

wVeem

Lk

**

*.

00 (5.13)

Here the average value of suffices K and L appearing in this equation is taken as

the diameter of the ith category of particles. Thus the particle number density of ith particle

size is obtained by dividing eq. (5.13) with Vi as given by eq. (5.3).

5.8. CASE STUDY

As a typical case of Rosin-Rammler distribution, the cumulative frequency curve

shown in fig.5.1 is used for generating the sub-volumes. This set of data is called as

“Primitive Cumulative Distribution” since 100 percent of solid particles are not accounted as

shown in Table-5.A.1 of Appendix-5.A. The frequency table is further processed for the

preparation of input, as shown in Table-5.A.2 of Appendix-5.A. From these data, ten

different classes are formed based on particle size. The different classes along with their

volumes, expansion angles and the radii at the nozzle exit plane and the axial height h are

given in Tables 5.D.1, 5.D.2 and 5.D.3 of Appendix 5.D.

5.9. RESULTS & DISCUSSIONS

The method of implementation of Rosin-Rammler distribution in an SRM plume is

explained and the geometric parameters of different particle sizes are derived. The three

geometric parameters, defining the limiting particle trajectories of each category of particles,

namely, the differential radii at the nozzle exit plane and at an axial height of ‘h’ along the

plume axis and its differential expansion angle are derived. For a typical input parameter

combination of h=9.713m & a=6.709m, these geometric parameters are presented for

different volumes of plume namely, 20m3, 40m3 and 60m3 in Tables 5.D.1, 5.D.2 and 5.D.3

of Appendix-5.D.

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5.10. CONCLUSIONS

Rosin-Rammler distribution in the exhaust plume of SRM is modeled. The mass

fraction of solid particles given by the Rosin-Rammler distribution is mapped into volumes

using the CFD results of particle trajectories. The envelop of different sizes of particles are

approximated as frustum of cones based on the polynomial equations of limiting trajectories

of particles of different sizes. The particle density in any control volume is defined for

evaluating the scattering efficiency of solid particles. The geometrical parameters of the

resulting frustum of cones is derived and applied for a tested solid motor. The resulting

geometrical parameters of different particle sizes are presented for three cases as a

function of volume of the computational domain.